Optical Excitation of Surface

Plasmon Polaritons on Novel

Bigratings

Thomas James Constant

School of Physics

University of Exeter

A thesis submitted for the degree of

Doctor of Philosophy

March 2013

Optical Excitation of Surface Plasmon

Polaritons on Novel Bigratings

Submitted by Thomas James Constant to the University of Exeter as a thesis for

the degree of Doctor of Philosophy in Physics

2013

This thesis is available for Library use on the understanding that it is copyright

material and that no quotation from the thesis may be published without proper

acknowledgement.

I certify that all material in this thesis which is not my own work has been identiﬁed

and that no material has previously been submitted and approved for the award of a

degree by this or any other University.

Thomas James Constant

2013

For my mum, Lel, and my sister, Emma.

Abstract

This thesis details original experimental investigations in to the interaction of light with

the mobile electrons at the surface of metallic diﬀraction gratings. The gratings used in

this work to support the resultant trapped surface waves (surface plasmon polaritons),

may be divided into two classes: ‘crossed’ bigratings and ‘zigzag’ gratings.

Crossed bigratings are composed of two diﬀraction gratings formed of periodic

grooves in a metal surface, which are crossed at an angle relative to one another. While

crossed bigratings have been studied previously, this work focuses on symmetries which

have received comparatively little attention in the literature. The gratings explored in

this work possesses two diﬀerent underlying Bravais lattices: rectangular and oblique.

Control over the surface plasmon polariton (SPP) dispersion on a rectangular

bigrating is demonstrated by the deepening of one of the two constituent gratings.

The resulting change in the diﬀraction eﬃciency of the surface waves leads to large

SPP band-gaps in one direction across the grating, leaving the SPP propagation in the

orthogonal direction largely unperturbed. This provides a mechanism to design surfaces

that support highly anisotropic propagation of SPPs.

SPPs on the oblique grating are found to mediate polarisation conversion of the

incident light ﬁeld. Additionally, the SPP band-gaps that form on such a surface are

shown to not necessarily occur at the Brillouin Zone boundaries of this lattice, as the

BZ boundary for an oblique lattice is not a continuous contour of high-symmetry points.

The second class of diﬀraction grating investigated in this thesis is the new zigzag

grating geometry. This grating is formed of sub-wavelength (non-diﬀracting) grooves

that are ‘zigzagged’ along their length to provide a diﬀractive periodicity for visible

frequency radiation. The excitation and propagation of SPPs on such gratings is

investigated and found to be highly polarisation selective.

The ﬁrst type of zigzag grating investigated possesses a single mirror plane. SPP

excitation to found to be dependant on which diﬀracted order of SPP is under polarised

illumination. The formation of SPP band-gaps is also investigated, ﬁnding that the

band-gap at the ﬁrst Brillouin Zone boundary is forbidden by the grating’s symmetry.

The ﬁnal grating considered is a zigzag grating which possesses no mirror symmetry.

Using this grating, it is demonstrated that any polarisation of incident light may

resonantly drive the same SPP modes. SPP propagation on this grating is found to be

forbidden in all directions for a range of frequencies, forming a full SPP band-gap.

Acknowledgements

First and foremost, I would like to thank my supervisors, Professor Roy Sambles and

Professor Alastair Hibbins. Roy has been a constant inspiration to me throughout my

research, and has kept me motivated and excited in all that I have done. The level of

care and diligence he aﬀords his students is second to none. Thank you Roy. In Alastair,

I have gained not just a respected colleague, but also a close friend. In the early days of

my supervisory meetings, Al was always there to step in and say “Hang on, Roy, this

isn’t trivial.”, for which I will be forever grateful. I’ve also enjoyed many fruitful hours

with Al at conferences, and many less fruitful hours down the pub. Thank you Al.

Many, many members of the Electromagnetic Materials Group at Exeter have helped

me throughout my Ph. D. None more so than Matt Lockyear, without whose advice and

support I most certainly wouldn’t have made it this far. Matt and I was fortunate enough

to be part of RAD crew during it’s golden age, a rag-tag bunch of physicist-surfers who

had some rather fantastic adventures.

Ian Hooper is Exeter’s resident grating expert, and I am indebted to him for countless

discussions, arguments and comments on the physics of the diﬀraction grating. Currently,

our running pool score stands at 97-65 to Hoops, and by the time anyone reads this I

am certain he will have reached 100 and I will have had to buy us a holiday.

In the ﬁrst few years of my work, I was lucky enough to have Professor Alex

Savchenko as my academic mentor. Alex helped me a great deal when I was struggling,

and was one of the most down to earth, empathic people I’ve ever met. Even after

his untimely death in 2011, Alex’s legacy continues to have an important eﬀect on my

research. Alex’s winning of the funding for Exeter’s graphene center brought reliable

nano-fabrication technology and immeasurably skilled physicists to Exeter, and in doing

so allowed me to create the samples presented in this thesis.

I must thank Dave Anderson for his supreme skill at electron beam lithography and

his remarkably patient and helpful attitude in dealing with me. A lot of the samples

used in this work simply could not have been made without Dave’s help.

Plenty of the other academics have been great fun to work with at Exeter. Pete

Vukusic has been a great help to me, providing me in the ﬁnal months with some

funding to develop the scatterometry kit further. Bill Barnes has always provided

thought provoking questions, and the back-to-basics sessions he has run helped me

lots with, well, the not-so-basics of our ﬁeld. Euan Hendry has also provided useful

discussions, and has also wiped me out many times in poker. A four hour chat we had

about Fourier series in optics, whilst sharing some beers in the baggage compartment of

a train from London to Exeter, still helps me heavily in the physical reasoning in this

thesis.

The technical workshop have been fantastic in my time at Exeter, making some

remarkable devices and some less remarkable Bar stands. Originally my go-to man was

Pete Cann, and after he left us to go enjoy Australia and then later family, the burden

then fell to Nick Cole. They are both fantastic technicians and have made many of

the results in the thesis possible by providing, or sometimes just ﬁxing, vital kit. The

other workshop boys have also been great. John, Kev and Adam have all helped me

greatly, be it helping with orders (John), making things Nick can’t (Kev), or just yelling

instructions at me as our star defender on the post-grad football team (Adam!). In the

techinical areas, I can’t forget Chris Forrest, who is often a lifesaver with all things IT.

Team Basement, as the dwellers of our dankest subterranean lab named ourselves,

were constant fun. Ed Stone, for sharing and increasing my love for R, introducing me to

Linux and providing some of the most memorable moments of these past years, I thank

you! Earthquake testing Ed, though ill-advised, is possibly one of the funniest things

you’ll ever see. Caroline Pouya has managed to put up with me now for close to nine

years. How she’s managed that is completely beyond me, and despite such prolonged

exposure to me, she has become one of my closest friends. Later, we were joined by Alﬁe,

Nixon and in the ﬁnal days Luke and Tim, who all took the challenge of living without

daylight and the constant fear of asphyxiation, electrocution or asbestos poisoning with

the good humour we’ve come to expect from Team Basement. I’d also like to thank

Ed and Alﬁe particularity for listening to my ideas about using scatterometry to map

iso-frequency contours, and helping me develop the experiential technique which is used

heavily in this thesis. Chris Holmes was also there, but we shall get to him later.

The microwave kids in G31 have also been wonderful friends. Celia Butler has always

been there for a chat, and I certainly now know more about Brownies and Gambia

than I ever intended to. Mel Taylor and I did some summer project work together just

prior to the Ph. D., and we had some great fun. Later Al, Simon, Panda, Liz, Laura,

Ben and Ruth joined us, and they’re all been fantastic friends. Another microwave

bunny, Helen Rance is (quite by accident) my oldest friend at Exeter. We’ve known

each other for so long, and through so many diﬀerent experiences, that to attempt a

decent acknowledgement is an exercise in futility.

For the majority of my Ph. D., I lived in Horseguards parade with Steve Hubbard,

Matt Biginton, Ciar´an Stewart, Chris Holmes and Pete Hale. Stormguards, as we

aﬀectionately named it, was not just a house. It was an exclusive gentleman’s club, a

brewery, a philosophical forum, a base of operations, a darts club, a support group, a

fraternity and a state of mind. The golden age of the Stormers saw us invent the amateur

beer festival, the glove game, tea-darts, Nakatomi Plaza, develop an appreciation of ﬁne

whisky and much much more. I think it’s fair to say that the two key players (other

than myself) in the development of Storm-Culture was Pete and Chris; so Stormers, I

thank you especially.

Plenty of other physicists have been important to me throughout this work. To list

a few, I must mention Nat, Babs, Laureline, Tom D, and not forgetting the Russian

contingent in Tim, Ivan, and the rest of the quantum group.

I have been fortunate to have many great friends outside of the department over the

past 8 years. Holly Keatings and I met on the second day of uni, and have together

sampled almost every restaurant in Exeter. My other house-mates of that era were

Ashley, Rob and Jon. It was my great privilege to see Lash marry a fellow Stormer,

Steve, after introducing them at our house while making name badges.

For many years I competed for the university as part of the fencing club. On the

Men’s 1

team I met Graham Heydon and and James Parker, who are now two of my

best friends. The adventures and trouble we got ourselves in make any attempt to play

‘I have never’ with these guys total suicide. I should also mention Pippa, otherwise

she’ll get in a huﬀ.

My Aunt, Terri and my Uncle, Joe have let me live with them for the past 5 months,

and have let me escape the city to be able to write this thesis. They have made me feel

exceptionally welcome and tremendously loved during my stay. I’m lucky enough to say

that this aﬀection and kindness isn’t unique to this period of writing, but is the norm

for my entire life. Thank you, Terri and Joe.

Finally I must thank Emma and Mum. My little sister Em has always been

encouraging and interested in what I’ve been doing, and as she’s grown up has become

one of my best friends. My mum, Lel, apparently used to ﬂash brightly coloured cards

at me while I was a baby in my cot, hoping to somehow induce intelligence. I’m not

sure that worked, but her wish, motivation and encouragement for me to succeed at

anything I put my mind to is pretty much a running theme of my entire life. It is

certainly why I get to be writing this today. I love them both very much.

Contents

Contents vi

List of Figures ix

1 Introduction 1

1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Scope and Outline of This Work . . . . . . . . . . . . . . . . . . . . . . 4

2 Background Theory 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Surface Polaritons on Planar Surfaces . . . . . . . . . . . . . . . . . . . 7

2.3 Dispersion of SPPs on Planar Surfaces . . . . . . . . . . . . . . . . . . . 11

2.3.1 Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Propagation Length . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Surface Plasmon Polaritons on Diﬀraction Gratings . . . . . . . . . . . . 16

2.4.1 Diﬀraction Gratings . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Coupling Surface Plasmons to Light by Coherent Scattering . . . 18

2.4.3 Surface Plasmon Polaritons on Bigratings . . . . . . . . . . . . . 23

2.4.4 Coupling Strength of Light to SPPs on Gratings . . . . . . . . . 25

2.4.5 Plasmonic Band Gaps . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.5.1 Coupling Light to Band Edges . . . . . . . . . . . . . . 29

2.4.6 Polarisation Conversion . . . . . . . . . . . . . . . . . . . . . . . 31

3 Theoretical Methods 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 The Diﬀerential Method of Chandezon et al. . . . . . . . . . . . . . . . . 34

3.3 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.1 Solving Maxwell’s Equations in the Mesh . . . . . . . . . . . . . 36

3.3.2 Adaptive Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.3 Solution Evaluation and Convergence . . . . . . . . . . . . . . . 37

CONTENTS

3.3.4 Model Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 37

3.3.5 Floquet Type Excitation . . . . . . . . . . . . . . . . . . . . . . . 39

4 Experimental Methodology 41

4.1 Sample Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . 41

4.1.2 Thermal Evaporation . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.3 Template Stripping . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1.4 Polymer Replication . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Angle Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.1 Monochromator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.2 Fixed Wavelength Scans . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.3 Embedded Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Iso-Frequency Contour Measurement Using Scatterometry . . . . . . . 49

4.3.1 Scatterometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.3 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.4 The Role of Polarisation . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.5 Corrections and Momentum-Space Deformation . . . . . . . . . . 55

4.3.6 Example Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5 Optical Response of Metallic Rectangular Bigratings 58

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 The Rectangular Bigrating . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.3 Dispersion of SPPs on Rectangular Bigratings . . . . . . . . . . . . . . . 60

5.3.1 Scattering Components on Lamellar Gratings . . . . . . . . . . . 60

5.3.2 Experimental Mapping of SPP Dispersion . . . . . . . . . . . . . 62

5.3.3 Band Gap Observation Using Scatterometry . . . . . . . . . . . . 66

5.4 Polarisation Conversion on Rectangular Symmetry . . . . . . . . . . . . 68

5.5 Controlling SPP Anisotropy Using Rectangular Bigratings . . . . . . . . 71

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Optical Response of Metallic Oblique Bigratings 77

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 The Oblique Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.3 Coupling of Light and SPP Mode Interaction on Oblique Gratings . . . 80

6.4 Polarisation Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.5 SPPs at the BZ Boundary of Oblique Bigratings . . . . . . . . . . . . . 85

6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

vii

CONTENTS

7 Optical Response of Metallic Zigzag Bigratings 92

7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7.2 The Zigzag Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.3 The Coupling of Plane Polarised Light to SPPs on Zigzag Gratings . . . 94

7.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.3.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.3.3 Transverse Electric Coupling . . . . . . . . . . . . . . . . . . . . 100

7.3.4 Transverse Magnetic Coupling . . . . . . . . . . . . . . . . . . . 104

7.4 Band Structure of SPPs on a Zigzag Grating . . . . . . . . . . . . . . . 107

7.4.1 Band-Gaps at k

= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.4.2 Band-Gaps at the First Brillouin Zone Boundary . . . . . . . . . 112

7.5 Anisotropic Propagation of SPP Modes for Self Collimation . . . . . . . 115

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

8 Optical Response of Asymmetric Zigzag Bigratings 122

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2 The Asymmetric Zigzag Grating . . . . . . . . . . . . . . . . . . . . . . 123

8.3

The Coupling of Polarised Light to SPPs on an Asymmetric Zigzag Grating

126

8.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.4 Band Structure of SPPs on an Asymmetric Zigzag Grating . . . . . . . 134

8.4.1 Band Gaps at k

= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.4.2 Band Gaps at the 1

BZ Boundary . . . . . . . . . . . . . . . . 136

8.4.2.1 Coupling of Light to the Band Edges . . . . . . . . . . 138

8.4.2.2 A Full Surface Plasmon Band Gap . . . . . . . . . . . . 141

8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

9 Conclusions 145

9.1 Summary of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Publications 150

References 152

viii

List of Figures

1.1

The four types of grating geometry that are investigated in this thesis:

The rectangular bigrating, the oblique bigrating, the zigzag grating, and

the asymmetric zigzag grating. . . . . . . . . . . . . . . . . . . . . . . . 4

2.1

A schematic representation of propagating electromagnetic ﬁelds at an

interface between two materials. . . . . . . . . . . . . . . . . . . . . . . 9

2.2

The Drude model for silver calculated with

= 1

Hz, γ

1.4 × 10

Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

The dispersion of a surface plasmon polariton on a planar ﬁlm approxi-

mated with the Drude mode. . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4

Diagram of the electric ﬁeld vectors of a SPP at the interface between a

metal and a dielectric and the exponential decay of the

component

away from the surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 The coordinate system used for a simple grating in the conical mount. . 17

2.6

The allowed real momentum states for light incident on three diﬀraction

gratings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7

Two dispersion diagrams illustrating in-plane diﬀractive coupling to

the zero-order SPP in the extended zone scheme, and the higher-order

diﬀracted SPPs couling to zero-order light in the reduced zone scheme . 21

2.8

3D plots demonstrating intersections of the plane of incidence with a

scattered SPP cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.9

The reciprocal space map for an oblique bigrating with grating vectors

and k

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.10

An example spectrum for a simple sinusoidal silver grating, calculated

using the Chandezon method. . . . . . . . . . . . . . . . . . . . . . . . . 25

2.11 Cartoon of two possible SPP standing waves on a sinusoidal grating. . . 27

2.12

The dispersions and band gaps of a SPP on a grating with one Fourier

Harmonic and on a grating with the ﬁrst two Fourier Harmonics. . . . . 28

LIST OF FIGURES

2.13

Sketches of diﬀerent grating proﬁles determined by the relative phases

of the

and 2

components, and the resulting coupling of the light to

the band-edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1

A graph showing the convergence of a typical FEM model as a function

of adaptive passes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Schematic of the diﬀerent boundaries used in the FEM model. . . . . . 39

4.1

Illustration of the fabrication method used for the production of diﬀraction

gratings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2

System for the measurement of the reﬂectivity of a sample as a function

of both wavelength, λ

and polar angle θ. . . . . . . . . . . . . . . . . . 47

4.3

System for the measurement of angular dependant reﬂectivity at a set

wavelength of λ

= 632.8 nm . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Exploded view of the sample mount for use with a glass hemisphere. . . 49

4.5

Three iso-frequency contours for the intersection of two SPP cones at

angular frequencies ω

, ω

and ω

. . . . . . . . . . . . . . . . . . . . . . 50

4.6

Experimental arrangement for the modiﬁed imaging scatterometry system.

4.7

Four raw images from the scatterometer for a wavelength of

= 650 nm

with diﬀerent polarisations. . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.8 The image correction for R(θ, φ) → R(k

sin θ, φ). . . . . . . . . . . . . . 56

4.9

The processing of a raw dataset from the scatterometer at a wavelength

of λ

= 650 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.1 The coordinate system for a rectangular grating. . . . . . . . . . . . . . 60

5.2

Three examples of unit cells for lamellar grating proﬁles, and the square

of the Fourier coeﬃcients, a

, for these groove proﬁles. . . . . . . . . . . 62

5.3

Scanning electron micrographs of the template-stripped rectangular bi-

grating in silver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.4

Reﬂectivity for diﬀerent polarisations and azimuthal angles for a rectan-

gular grating mapped as a function of (ω, k). . . . . . . . . . . . . . . . 64

5.5

A scattergram of a rectangular grating mapped to

-space for an illumi-

nation wavelength of λ

= 550 nm. . . . . . . . . . . . . . . . . . . . . . 66

5.6

Cartoon of the (

−

0) and (1

0) SPP iso-frequency contours at the BZ

boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.7

Plots showing the electric ﬁeld for the high energy and low energy SPP

standing waves occurring at the BZ boundary. . . . . . . . . . . . . . . . 68

5.8 Spectra of polarisation conversion on the rectangular grating at φ = 45

◦

. 70

LIST OF FIGURES

5.9

Diagram of the eﬀect on the surface proﬁle by increasing the depth

the rectangular bigrating. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.10

A sketch showing the expected iso-frequency contour deformation as

is changed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.11

Experimental iso-frequency contours for two rectangular bigratings at a

wavelength of λ

= 700 nm. . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.12

Experimentally obtained dispersion diagrams mapped from the reﬂectance

of rectangular bi-gratings with nominal depths of 40 nm and 80 nm . . . 75

6.1 Coordinate system for an oblique bigrating. . . . . . . . . . . . . . . . . 78

6.2 Example reciprocal lattices for an oblique grating. . . . . . . . . . . . . 79

6.3 SEM of an oblique grating master fabricated in a silicon wafer. . . . . . 80

6.4

The modelled and experimental iso-frequency surface for an oblique

bigrating illuminated with TM polarisation at λ

= 700 nm. . . . . . . . 81

6.5 The dispersion of modes on an oblique grating for φ = α = 75

◦

. . . . . . 83

6.6 Polarisation conservation and conversion mediated by out-of-plane scat-

tered SPPs on an oblique grating. . . . . . . . . . . . . . . . . . . . . . . 85

6.7

An experimentally obtained iso-frequency contour map of SPPs on an

oblique bigrating with illuminated with incident light of wavelength

= 650 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.8 Regions of interest from ﬁgure 6.7. . . . . . . . . . . . . . . . . . . . . . 87

6.9 Two possible primitive unit cells for an oblique lattice. . . . . . . . . . . 89

6.10

Diagram of the cancellation of a vector ﬁeld on a rectangular and oblique

BZ boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.11 Dispersion plots mapped using the reﬂectivity of an oblique grating. . . 91

7.1 The coordinate system of a zigzag grating. . . . . . . . . . . . . . . . . . 94

7.2

The surface normal components of electric ﬁeld vector for TM and TE

polarised light projected on a contour of the zigzag surface proﬁle. . . . 95

7.3 The magnitude of the surface normal electric ﬁeld in the x-direction for

TM and TE polarisations. . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.4 Schematic cartoon of light coupling to SPPs on a zigzag grating. . . . . 98

7.5 Scanning electron micrographs of various zigzag grating samples. . . . . 99

7.6 SPP dispersion on a zigzag grating measured using TE polarised light. . 100

7.7

Plot of TE and TM polarised light reﬂectivity from the grating as a

function of polar angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

7.8 The E ﬁeld plot of a SPP excited by TE radiation. . . . . . . . . . . . . 102

LIST OF FIGURES

7.9

SPP dispersion on a zigzag grating in glass measured using TE polarised

light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.10

An example experimental reﬂectivity plot for TE and TM polarised light

in the visible range at a ﬁxed polar angle of θ = 53

◦

. . . . . . . . . . . . 105

7.11

An example spectral plot and ﬁt at

= 53

◦

for the TM reﬂectivity

normalised to the TE reﬂectivity, and comparison to theory. . . . . . . . 106

7.12

SPP dispersion on a zigzag grating in glass measured using TM polarised

light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.13

An iso-frequency scattergram mapping

-space contours for a 600 nm

zigzag grating for an energy of 2.14 eV . . . . . . . . . . . . . . . . . . . 108

7.14

Cartoon of the two standing wave solutions for SPPs at

= 0, projected

onto the zigzag surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.15 The magnitude of electric ﬁeld for the SPP standing waves at k

= 0. . 110

7.16

Manipulation of the band-gap observed at

= 0 through the increasing

of zigzag amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.17

Modelled SPP dispersion around the intersection point of the

−

and

+2k

scattered SPPs meeting at the ﬁrst BZ boundary. . . . . . . . . . 112

7.18 Cartoon of the two standing wave solutions for SPPs at the 1

BZ. . . . 113

7.19

The magnitude of electric ﬁeld,

|E|

for the degenerate SPP standing

waves at the ﬁrst BZ. (xy plane) . . . . . . . . . . . . . . . . . . . . . . 114

7.20

The magnitude of electric ﬁeld for the degenerate SPP standing waves at

the ﬁrst BZ. (xy and xz plane) . . . . . . . . . . . . . . . . . . . . . . . 115

7.21

Experimentally obtained dispersion plots for a zigzag grating at

= 0

◦

and φ = 90

◦

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.22

Measured iso-frequency contours of a zigzag grating for a range of wave-

lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.23

The SPP iso-frequency contours mapped using imaging scatterometery

at λ

= 450 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.1 Coordinate system for the asymmetric zigzag grating. . . . . . . . . . . 124

8.2 Two possible unit cells for a zigzag grating: symmetric and asymmetric. 125

8.3

Scanning electron micrographs of an asymmetric zigzag silicon master

and the template stripped sample in silver. . . . . . . . . . . . . . . . . 125

8.4

A schematic of the electric ﬁeld lines in the grooves of an asymmetric

zigzag grating for two polarisation cases. . . . . . . . . . . . . . . . . . . 127

8.5

The piecewise function representing

for TE (

T E

) polarisation and

TM (E

T M

) polarisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xii

LIST OF FIGURES

8.6

Magnitude of the electric ﬁeld in region 1 and region 2, for the two

polarisation cases of TM and TE polarised light. . . . . . . . . . . . . . 130

8.7

The square magnitude of the Fourier coeﬃcients for

= 0

... ±

4 with an

oﬀset of δ = 0.42L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.8

The TE and TM reﬂectivity as a function of polar angle for an asymmetric

zigzag grating illuminated with a wavelength of λ

= 500 nm. . . . . . . 132

8.9

The dispersion of the SPP modes mapped as a function of TE and TM

reﬂectivity on an asymmetric zigzag grating. . . . . . . . . . . . . . . . . 133

8.10

The dispersion of the SPP modes mapped as a function of unpolarised

light reﬂectivity on an asymmetric zigzag grating. . . . . . . . . . . . . . 134

8.11

The iso-frequency contours at

= 550 nm, measured on a symmetric

zigzag and an asymmetric zigzag grating. . . . . . . . . . . . . . . . . . 135

8.12

The calculated eigenmodes for the crossing of the

−

and +2

scattered SPPs at the ﬁrst BZ for the cases of: a symmetric zigzag; a

symmetric, high-amplitude zigzag and; an asymmetric zigzag. . . . . . . 137

8.13

Field plots of

|E|

(colourplot) a for the low energy and high energy

standing waves at the ﬁrst BZ. . . . . . . . . . . . . . . . . . . . . . . . 138

8.14

Field plots of

|E|

(colourplot) and

(arrows) for the low energy and

high energy standing waves at the ﬁrst BZ. . . . . . . . . . . . . . . . . 139

8.15

Experimental reﬂectivity colour plots mapping the dispersion of the band

gap around the 1st BZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8.16

The low energy and high energy

-component of the SPP standing wave

electric ﬁeld and the incident ﬁeld across two unit cells of the asymmetric

zigzag grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.17

Iso-frequency contours of an asymmetric zigzag grating for a range of

wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

8.18

The dispersion of the SPP modes on an asymmetric zigzag grating for

φ = 0

◦

, and the scattergram for λ

= 450 nm. . . . . . . . . . . . . . . . 143

9.1

A possible experimental arrangement for the observation of collimated

surface waves using zigzag gratings. . . . . . . . . . . . . . . . . . . . . . 148

xiii

Chapter 1

Introduction

This thesis details original experimental investigations in to the interaction of light

with the mobile electrons at the surface of metallic diﬀraction gratings. The resulting

quantised surface waves, surface plasmon polaritons, have been investigated by optical

scientists for over a century, yet interest in the ﬁeld has never been higher, and progress

has never been faster [

]. This is due in large part to the relatively recent advances in

nano-fabrication techniques, which have allowed greater control over the surface wave

propagation and dispersion characteristics on metallic surfaces. This work endeavours

to extend the understanding of surface plasmon polaritons on diﬀraction gratings by

exploring their propagation along symmetries and structures that have not been reported

previously, and whose fabrication has only recently become achievable.

1.1 History

The phenomenon of diﬀraction from gratings has been reported by scientists since

Francis Hopkinson peered at a street lamp through a silk handkerchief in 1785 [

The rainbow that is seen due to the dispersion of light through such ﬁne structure was

recognised immediately as an incredibly useful optical property by Fraunhofer, and it

was not long before early diﬀraction gratings were being manufactured by scratching

ﬁne grooves in to the surfaces of glasses and metals.

In 1902, it was these early forays into the structuring of metals that led to the ﬁrst

recorded observations of surface plasmon polaritons’ interaction with light. Wood [

]

reported a bright band of enhanced reﬂection and a dark band of low reﬂection found

in the projected spectrum of a ruled speculum diﬀraction grating produced at John

Hopkins University [

]. Diﬀraction gratings produced by these early ruling engines were

ﬁnding a multitude of uses in physics at the time, particularly in the young ﬁeld of

spectroscopy, and an explanation of Wood’s spectral anomalies was vigorously sought.

1. Introduction

Lord Rayleigh explained the observed dark band in his theory of diﬀraction gratings

in 1907 [

] as the wavelength of incident light at which a diﬀracted light ray will start,

or cease, to propagate. The redistribution of available propagating energy at this

point created a reﬂectivity feature corresponding to one of the two Wood’s anomalies.

Thereafter this was referred to as the ‘Rayleigh anomaly’ and today this is also called

the ‘diﬀraction edge’ or ‘pseudo-critical edge’.

Wood’s bright band was left unexplained for nearly thirty years until Strong [

attempting to improve the reﬂective eﬃciency of a set of gratings by coating them with

diﬀerent types of metals, noticed the band shifted in energy depending on the metal

coating used. This dependence on the metal’s optical characteristics led Ugo Fano

[

–

], to the conclusion that this reﬂectivity anomaly was a signature of interactions

of light with a trapped surface wave, comprised in part by the metal’s conduction

electrons at the surface. He took the view that this represented a ‘zero-order’ waveguide

mode: a guided mode that could still exist localised to the surface when a hypothetical

supporting waveguide over-layer was decreased to zero thickness. Maxwell’s equations

provided Fano with a solution for such a wave, on the condition that one material was

a conductor and the other a dielectric

∗

. Today we call this interface mode a surface

plasmon polariton (SPP) and it is deﬁned as an electromagnetic surface wave bound to

the interface between a conductor and a dielectric.

While the ﬁrst recorded observation of SPPs was on a diﬀraction grating, SPPs

are not unique to these devices alone, and are generally found in systems where a

conducting/dielectric interface exists. Once Fano had underpinned the physical origin

of this surface wave, and in particular the momentum requirements to resonantly drive

it, other methods by which to excite SPP on metal surfaces began to emerge. In the

late 1950s, Ritchie [

] and Ferrell [

] both predicted the excitation of SPPs on thin,

ﬂat, metal ﬁlms with accelerated electrons, with Ferrell also predicting the SPPs could

decay back into light. This prediction was conﬁrmed experimentally by Steinmann [

]

two years later. Today, the use of electrons to excite SPPs has found application as a

very precise method of plasmonic investigation: electron energy loss spectroscopy [17].

In 1968, Otto [

] and simultaneously Kretschman & Raether [

] excited SPPs

using a prism geometry

†

. This technique used the evanescent ‘tail’ of light undergoing

total internal reﬂection in the prism to couple to SPPs on a metal/air interface, matching

the momentum of the SPP ﬁeld through the classical tunnelling of the evanescent wave

originating from inside the high-index material. Their techniques for prism coupling are

∗

Unbeknownst to Fano, similar work had been pursued by Zenneck [

] and also Sommerfeld [

] in

their research on the transmission of radio-waves across large distances using trapped surface waves

between seawater and air.

†

Turbadar [

] was the ﬁrst to excite this wave using a prism geometry, but unfortunately failed to

connect his work to that of Fano or his observed phenomena to SPPs.

1. Introduction

still used today, particularly in SPP based sensors which can investigate a diverse range

of analytes, including the detection of fake tequila [21] and other spirits [22].

Research in to SPPs on diﬀraction gratings continued in parallel with the develop-

ment of these other excitation methods, with their observation and dispersion being

investigated in the 1960s [

–

]. Research in the 1970s and 1980s focussed on the

eﬀect of deep grooves [

–

] and over-layer coatings [

], driven in part by the

development of holographic methods for diﬀraction grating manufacture [

]. This was

followed by work in the 1990s that explained the inﬂuence of the grating shape on the

resonances [

], SPP’s role in polarisation conversion [

] and the physical origin of

observed SPP band interactions [35].

With the advances in nano-fabrication techniques, the development of detailed

theoretical treatments for diﬀraction problems [

] and the increasing aﬀordability

and power of computers, increasingly complex diﬀraction grating geometries could be

explored. The combinations of all these factors also helped spawn the active research

ﬁeld of ‘plasmonics’, and further improvements in all these areas continue to be a driving

force behind the latest SPP research.

SPPs on gratings have already found many applications, and in some cases are

already integral to commercial products. With the energy of a SPP resonance being

highly sensitive to the local environment at the surface, SPPs have been used very

successfully in sensing applications [37–39]; these have included the in-situ monitoring

of chemical reactions such as catalytic conversion [

], non-contact determination of

surface structure [42, 43], and the measurement of optical constants of metals [44].

Coupled SPPs on gratings have also been used to enhance various optoelectronic

devices; improvements to photo-detectors [

], the enhancement of laser beams [

–

]

and the ability to improve the eﬃciency of solar cells [

–

] have all been reported in the

literature. Generation of radiation has also been achieved on metallic or semiconducting

gratings through SPP mediated second harmonic generation [

] or by SPPs stimulating

electron photo-emission leading to the production of terahertz scale radiation [56, 57].

Additionally, SPPs on gratings have been employed as optical elements including

colour ﬁlters [

], polarisation converters [

], and various surface optics for collimation

or achieving negative refraction [

]. They have also been used for the novel demon-

stration of enhanced transmission through hole arrays [

–

] and the manipulation of

Magneto-optical eﬀects such as Faraday and Kerr rotations [

]. The use of SPPs will

no doubt continue to ﬁnd applications in the most recent of technologies, particularly

with very recent demonstration of their excitation on graphene [67].

This brief review of the investigations and applications of SPPs on gratings leads us

to the present day and the theme of this thesis. In this work, unconventional gratings

are used to couple to SPPs, and the propagation and dispersion of SPPs on such gratings

1. Introduction

(a) Rectangular (b) Oblique

Figure 1.1: The four types of grating geometry that are investigated in this thesis: (a)

The rectangular bigrating in chapter 5, (b) the oblique bigrating in chapter 6, (c) the

zigzag grating in chapter 7 and (d) the asymmetric zigzag grating in chapter 8.

is investigated. The experiments presented here are done to explore the properties of

SPPs on novel types of grating, and do not attempt to ﬁt any particular application.

However, many possibilities for extensions or applications are possible, and these are

suggested in each chapter and summarised in the conclusions.

1.2 Scope and Outline of This Work

This thesis details experimental investigations into the propagation of SPPs on diﬀraction

gratings that possess novel structure or symmetries. Broadly it may be divided up in to

investigations of two types of diﬀraction grating; ‘crossed’ bigratings and a new type

of diﬀractive optical element: the ‘zigzag grating’. Since both these types of grating

possess two diﬀerent diﬀractive periods in their surface geometry, they may both be

considered a type of the larger family of metallic ‘bigratings’. There are four diﬀraction

grating geometries investigated in this work, and these are summarised in ﬁgure 1.1.

They are; the rectangular bigrating, the oblique bigrating, the ‘zigzag grating’ and the

‘asymmetric zigzag grating’.

The background theory of SPPs on both planar ﬁlms and on metallic diﬀraction

gratings is presented in chapter 2. This chapter covers the origin, coupling conditions

1. Introduction

and band-structure of SPPs on both planar and periodic surfaces. The methods by

which the optical response of the gratings under consideration have been calculated

theoretically are then explained in chapter 3.

Chapter 4 details the experimental methods used for the production and measure-

ments of gratings in this thesis. In addition to the standard experimental techniques

used, a new method by which to map the plasmonic analogue to the iso-frequency

‘Fermi-contours’ of the SPP band structure is presented. These iso-frequency contours

have been recorded using imaging scatterometry and this new technique, developed as

part of this body of work, is used extensively throughout this thesis.

Chapter 5 presents some experimental observations on the excited SPPs supported by

rectangular bigratings. These are gratings formed of two diﬀraction gratings of diﬀerent

pitch, crossed relative to each other at an angle of 90

◦

. The dispersion of these modes

on the surface and the formation of standing surface-wave states are experimentally

recorded and matched to a theoretical model. It is found that by deepening the grooves

of one of the constituent diﬀraction gratings, the propagation of SPPs along the surface

becomes highly anisotropic. Control over this eﬀective mode index in diﬀerent directions

along the surface could ﬁnd application in surface-wave optics devices.

The work on ‘crossed’ bigratings continues in chapter 6, with the experimental

investigation of SPPs on a bigrating with the reduced symmetry of an oblique lattice.

The dispersion and scattering mechanisms on a fabricated oblique grating are recorded

experimentally and explained. SPP mediated polarisation conversion is also observed

on these gratings, as the scattered surface ﬁelds propagate along a surface of no mirror

symmetry. The lack of symmetry on such a grating leads to the observation of SPP

band gaps not forming at the Brillioun Zone (BZ) boundaries, and a general discussion

of the oblique symmetry constraints on SPPs is oﬀered to explain why this is so.

Chapter 7 introduces a new type of diﬀraction grating, the ‘zigzag grating’. This

grating uses sub-wavelength structure in one direction to introduce an diﬀractive

periodicity in the orthogonal direction. It is found, experientially and theoretically, that

even-order diﬀracted ﬁelds only couple to SPPs for one linear polarisation of light, while

odd-order diﬀracted ﬁelds only couple to the other, orthogonally polarised light. Further,

it is shown that SPP band gaps are forbidden at the ﬁrst BZ boundary by the symmetry

of the zigzag surface. Finally in this chapter, it is shown that the sub-wavelength grooves

on such a grating can lead to highly anisotropic SPP propagation. This anisotropy leads

to SPP propagation at certain frequencies in only one single direction, irrespective of

excitation angle. When combined with the lack of band gaps on such a symmetry, this

makes these gratings excellent candidates for surface wave collimation devices.

The ﬁnal experimental results of the thesis are presented in chapter 8, which extends

the zigzag grating geometry to one with reduced symmetry. This relaxes the polarisation

1. Introduction

conditions of the previous zigzag grating, leading to any SPP being coupled to with

any incident polarisation of light, a result which could prove relevant to improving

the eﬃciency of many plasmonic devices. The band structure of the SPPs supported

by this grating is also experimentally investigated and the coupling to the diﬀerent

standing wave states by light which occur at the ﬁrst BZ boundary is explored. The SPP

anisotropy previously found for a zigzag grating is also found in this new asymmetric

zigzag grating case, and combined with the large band gaps which form at the ﬁrst BZ

boundary, the grating is shown to support a full plasmonic band gap, for which SPP

propagation is forbidden in all directions.

The thesis is concluded in chapter 9 with a summary of the ﬁndings and suggested

future research which could extend the ﬁndings and applications of this work.

Chapter 2

Background Theory

2.1 Introduction

The interaction of light with metals has long proven to be a reliable path to striking

optical eﬀects. The Lycurgus cup is an example of a 4th century plasmonic device;

a glass Roman goblet which uses colloidal gold and silver nano-particles to achieve a

dull green surface reﬂection, but glows blood red when illuminated from the inside

[68]. Resonant interaction between light and the electrons in these suspended metallic

nano-particles provide the mechanism for this vivid eﬀect, the same eﬀect found in the

majority of stained glass windows.

The topic of this thesis also concerns the coupling of light to electrons, speciﬁcally

the mobile electrons found at the interface between dielectric and a conductor. Surface

plasmon polaritons are electromagnetic surface waves coupled strongly to the longitudinal

oscillations of this free electron plasma at the interface. They are quantised surface

waves that propagate along the interface, evanescently decaying in the normal direction.

This chapter introduces the background electromagnetic theory of these surface waves

and their interaction with light.

Sections 2.2 and 2.3 deal with the origin and characteristics of surface plasmon

polaritons propagating along a ﬂat interface. Metallic diﬀraction gratings provide a

mechanism by which to couple light to these surface waves. This coupling, and the

constraints placed upon surface waves on a such periodic surfaces, are detailed in section

2.4.

2.2 Surface Polaritons on Planar Surfaces

Surface plasmon polaritons (SPPs) can be categorized as a member of a larger family of

surface waves, broadly named ‘surface polaritons’. Surface polaritons are electromagnetic

2. Background Theory

surface waves coupled strongly to an elementary excitation and bound evanescently to

the interface between two media. When a photon couples to the longitudinal oscillations

of the free-electron plasma on metal surfaces, the resulting surface polariton is called a

surface plasmon polariton. Surface polaritons also couple with surface lattice vibrations

(surface phonon polaritons) [

], surface electron-hole pairs in semiconductors (surface

exciton polaritons) [

] and collective excitations of surface electron spin (surface magnon

polaritons) [

]. More recently, the manufacture of resonant sub-wavelength structures,

or ‘metamaterials’, has allowed the design of surfaces with tailored resonances other

than such elementary excitations, which may couple strongly to photons and produce

surface polaritons [

]. The resulting surface polaritons are often referred to as a

‘spoof’ surface plasmon polaritons.

∗

Solving Maxwell’s equations for an evanescently bound electromagnetic wave at an

interface leads to a similar relationship between energy and momentum for all surface

polaritons. The physical origin of the diﬀerent types of surface excitations and their

coupling to light is expressed in terms of the dielectric and magnetic functions which

are frequency dependent and complex in general.

Fano [

] derived the dispersion relation for a trapped surface wave by ﬁrst consider-

ing light propagating in a glass plate of ﬁnite thickness, bounded by semi-inﬁnite vacuum

and metal half-spaces. In a simple waveguide such as this, total internal reﬂections

prevent the propagating light from escaping into the bounding medium. The necessary

continuity of electric ﬁeld across a boundary for light undergoing total internal reﬂection

also requires non-propagating ‘evanescent’ waves extending into the bounding media,

which do not transfer any power. The question as posed by Fano [

] was then: ‘Is

there left any proper value when the thickness vanishes?’. As the thickness of the glass

plate tends to zero, there is indeed still a valid solution to Maxwell’s equations for a

wave travelling along the surface, evanescently bound in the normal direction; a surface

polariton. In these terms, a surface polariton can be thought of as the lowest-order

waveguide mode.

The dispersion of the surface polariton considered in this thesis, the surface plasmon

polariton on a metal/dielectric interface, allows us to simplify the problem to that of

isotropic, non-magnetic media. The derivation of this relationship [

] is presented

below, and is also valid for other surface polaritons in isotropic non-magnetic media,

such as surface phonon polaritons.

A schematic diagram of the system is shown in ﬁgure 2.1. Light illuminates a

planar surface between two media of permittivities

and

in the

plane, which we

deﬁne to be the plane of incidence. The wavevector of the light in the

medium is

∗

although the inclusion of ‘plasmon’ is not strictly correct as the photon is not coupled to the free

electron plasma oscillations but more usually a sub-wavelength cavity resonance.

2. Background Theory

ˆx

ˆz

Figure 2.1: A schematic representation of propagating electromagnetic ﬁelds at an

interface between two materials. The green rays illustrate the direction of the waves,

which are coincident with the wavevectors,

. The electric ﬁeld is polarised in the

plane, represented by black arrows. The optical response of the two media are

characterised by their permativities,

and

. The

ˆy

-direction is out of the page and

the position z = 0 is at the interface.

ˆx

ˆz ≡

[

, k

], with no component in the

ˆy

-direction (out of the

page). The light is then reﬂected back into medium 1 and refracted into medium 2.

For light polarised with the electric ﬁeld parallel to the plane of incidence (Transverse

Magnetic, or TM polarised) the plane waves can be described as,

= [E

, 0, E

] e

i(k

·x+k

·z)

−iωt

= [0, H

, 0] e

i(k

·x+k

·z)

−iωt

where

is the angular frequency of the light,

is time and

is a subscript indicating

in which medium the ﬁeld is propagating.

is the component of magnetic ﬁeld in

the

ˆy

-direction and

x,z

is the component of electric ﬁeld in the

ˆx

and

ˆz

directions,

respectively. Since the surface polariton is a trapped surface wave, we set the incident

wave to zero. Setting

to zero, we are left with two sets of ﬁelds (electric and magnetic)

for the half-spaces above and below the interface,

z > 0







= [E

, 0, E

] e

i(k

x+k

−iωt

= [0, H

, 0] e

i(k

x+k

−iωt

(2.1)

z < 0







= [E

, 0, E

] e

i(k

x−k

−iωt

= [0, H

, 0] e

i(k

x−k

−iωt

(2.2)

We may combine the equations for electric and magnetic ﬁeld in the two half-spaces

2. Background Theory

using Amp`ere’s law,

∇ × H

= ε

∂E

∂t

where

is the permittivity in medium

. The relationships between the tangential

electric and transverse magnetic ﬁelds in each material are then,

= +ωε

(2.3)

= −ωε

(2.4)

Having obtained expressions for the electromagnetic ﬁelds in both media, we must now

consider the continuity of these ﬁelds over the boundary. At the interface the boundary

conditions for the electromagnetic waves are expressed as,

= E

(2.5)

= H

(2.6)

= ε

(2.7)

These are the conditions that tangential electric ﬁelds, transverse magnetic ﬁelds and

the normal component of the electric displacement vector (

) must be

continuous across the boundary at

= 0. The continuity of the tangential electric ﬁeld

(equation 2.5) allows us to combine equations 2.3 and 2.4,

= 0

and continuity of transverse magnetic ﬁeld (equation 2.6) then yields

∗

= 0 (2.8)

Finally, to obtain a relationship in terms of the momentum of the surface wave in the

ˆx

direction (

), we consider the conservation of momentum in both regions. Conservation

of tangential momentum requires that

and so the expression for total

conserved momentum is,

+ k

= ε





(2.9)

Obtaining expressions for

and

using this conservation of momentum expression

and combining them with equation 2.8, we ﬁnd the dispersion relation for the surface

∗

At this point, we also employ some mathematical sleight of hand to substitute the permittivity,

, for the relative permittivity (

/ε

) by cancelling the common factor of

and

. For clarity

we redeﬁne ε

as the relative permittivity for the remainder of this thesis.

2. Background Theory

polariton,

+ ε

(2.10)

This equation relates the angular frequency of the ﬁeld,

to the wavevector along

the surface,

. The energy and momentum of the surface mode is related to these

quantities respectively by a factor of the reduced Planck’s constant,

. Because of this,

it is also often referred to as the energy-momentum relation.

The ﬁnal observation to be made in regards to the surface polariton dispersion is

the set of

and

values which can support a bound surface wave. The continuity

of normal electric displacement was mentioned brieﬂy in equation 2.7, and we shall

re-print it here.

= ε

This boundary condition states that the normal electric displacement must remain

continuous across the interface. However, from the plane wave equation sets (equations

2.1–2.2) it is apparent that

is always 180

◦

out-of-phase with

will always

be of opposite sign to E

. This is intuitive, as induced surface charge at the interface

will naturally cause the electric ﬁeld to extend into each surrounding media in opposite

directions. To satisfy the boundary condition, and so support surface polaritons on such

a surface, one material must be capable of ‘inverting’ the electric displacement resulting

in the condition that the permitivities ε

and ε

must also be of opposite signs.

2.3 Dispersion of SPPs on Planar Surfaces

The optical response of a metal is characterized by the metal’s frequency dependent

permittivity, also called the ‘dielectric function’ of the material. This optical response

function is dominated by two characteristics of metals; the fact that conduction electrons

are free to move within the bulk of the material, and the presence of inter-band transitions

of the valence electrons in the atomic orbitals. In the visible region, conduction electron

behaviour is ballistic, oscillating and interacting with light many times before scattering

from the crystal or other electrons. The inter-band transitions for a noble metal such

as silver occur at the edges of the visible spectrum and so the dominant eﬀect in the

appearance of silver is due mostly to the strong interaction of the free conduction

electrons with optical ﬁelds. At the surface of the metal, electromagnetic ﬁelds coupled

to longitudinal oscillations of this free-electron plasma form together a surface polariton:

the surface plasmon polariton.

The dispersion of a surface plasmon polariton takes the same mathematical form as

equation 2.10 with one dielectric function representing the frequency dependent metal,

2. Background Theory

(

), and the other representing a bounding non-conducting crystal with a dielectric

constant, ε

(ω)

+ ε

(ω)

(2.11)

A useful approximation for the metal dielectric function was presented by Drude

[

]. Drude considered free electrons travelling in a classical manner, scattering from the

ionic lattice with a characteristic average scattering rate. Electron-electron scattering is

disregarded and the permittivity is expressed as,

(ω) = ε

∞

−

+ iωγ

(2.12)

where

is the angular frequency of the light,

is the average rate of collision of the

free conduction electrons with the lattice, and

∞

is the response of the ionic lattice,

which is frequency independent and for metals in the visible domain equals 1. The

‘plasma frequency’ of the metal,

, is the natural frequency of oscillation of the bulk

conduction electrons.

An example of the permittivity of silver using the Drude model is show in ﬁgure 2.2.

It shows, for all visible regime wavelengths, that the real part of the silver permittivity

is negative and there is a non-zero imaginary component. This is generally true for a

wide range of metals at visible wavelengths. This negative permittivity of metals in the

visible domain, when bounded by a dielectric with a positive real permittivity, satisﬁes

the boundary conditions (equation 2.7) for a surface plasmon polariton. Substituting

2.12 in to the dispersion relation for a surface plasmon polariton (equation 2.11), we

obtain a fair approximation of the dispersion relation for a surface plasmon polariton on

a planar metal ﬁlm. This is plotted in ﬁgure 2.3. Figure 2.3 shows the dispersion of a

surface plasmon polariton on a ﬂat interface between silver and air. Plotted on the same

scale is a line representing a grazing photon along the surface, the ‘light-line’. Notice

that the light line and the surface plasmon polariton dispersion line do not cross. There

is no solution at which the energy and momentum of free-space light is equal to that

of the surface plasmon polariton. Since for a given energy of light, the light possesses

insuﬃcient momentum to match that of an SPP, the conclusion is that free-space light

incident on a ﬂat metallic surface cannot resonantly drive a surface plasmon polariton.

2.3.1 Penetration Depth

An excited SPP at the interface between a conductor and dielectric will possess elec-

tromagnetic ﬁelds which decay exponentially into both bounding media. A useful

measure of this decay is the penetration depth,

, which is the distance at which the

2. Background Theory

400 500 600 700 800 900

-35 -30 -25 -20 -15 -10

0.5 1 1.5 2 2.5 3

wavelength (nm)

−18.6

0.92

Figure 2.2: The Drude model calculated with

= 1

Hz, γ

= 1

As an example, the wavelength for a HeNe laser (632.8 nm) is highlighted, showing the

permittivity of silver at this wavelength to be ε(632.8 nm) ≈ −18.6 + 0.92i

ﬁeld amplitude has decreased to 1

of it’s maximum value. Momentum conservation

for a SPP (where

> ε

) gives an expression

− k

(equation 2.9

simpliﬁed.), which leads to the conclusion that

for SPPs must be purely imaginary.

Substituting an imaginary

into an expression for the electric ﬁeld at the surface

gives

∗

= [E

, 0, E

] e

−k

(2.13)

which is an expression for an electric ﬁeld which does indeed decay exponentially into

the two bounding media, and travels along the surface in the

direction. The value of

for which

falls to

−1

of the maximum value is then 1

. Substituting equation

2.10 into equation 2.9 we ﬁnd that the expression for k

is given by,

= ±k

−



+ ε



= ±k

+ ε

(2.14)

∗

the time dependent term, e

−iωt

has been omitted for clarity.

2. Background Theory

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0 2×10

4×10

6×10

8×10

0 2×10

6×10

10×10

√

Figure 2.3: The dispersion of a surface plasmon polariton on a planar ﬁlm approximated

with the Drude model with ω

= 1.32 × 10

Hz, γ = 1.4 × 10

Hz.

We may simplify this expression for the case under consideration with medium

= 1

being a non-absorbing dielectric,

(

)

0 and

(

) = 0, and medium

= 2 a lossy

metal,

(

)

0 and

(

)

0. In the case where the metal is highly conducting

we also have the considerations

|Re

(

)

| 

1 and

|Re

(

)

|  Im

(

). Under these

conditions, equation 2.14 simpliﬁes to,

= ±k

Re(ε

)

Re(ε

)

(2.15)

The penetration depth is then,

= λ

2π

|Re(ε

)

(2.16)

For a typical planar silver surface in air with

= 2

π/λ

= 2

π/

632

8nm, the penetration

depth for the SPP into the air is calculated, using equation 2.16, to be

≈

415 nm and

in to the metal

≈

24 nm. The ﬁeld at the surface and this associated exponential

2. Background Theory

++ ++−−

(a)

(b)

Figure 2.4: (a) Diagram of the electric ﬁeld vectors of a SPP at the interface between a

metal and a dielectric. (b) The exponential decay of the

component away from the

surface, with a maximum amplitude at

= 0 and the diﬀerent penetration lengths for

the two materials shown.

decay of the SPP ﬁelds in the

direction is shown diagrammatically in ﬁgure 2.4 for an

SPP in the visible regime.

In the limit of a perfect conductor, where

→ −∞

, equation 2.16 shows that the

penetration depth into the metal

→

0 and the penetration depth into the dielectric

tends to

→ ∞

. With no ﬁeld in the metal and the ‘decay length’ in

never actually

decaying in the dielectric medium, the result is a grazing plane wave and no localisation

of the ﬁeld to the surface can be said to exist. This is why the excitation of SPP

waves in the microwave regime is not observed on unaltered planar surfaces, as at these

frequencies the metals approximate a perfect conductor and so the vanishingly-small

localisation of the bound surface modes leads to the SPP resembling a grazing photon.

2.3.2 Propagation Length

By similar considerations of the surface electric ﬁeld, we deﬁne the propagation length

of the SPP along the surface as the length at which the ﬁeld intensity has fallen to

−1

of its maximum value. While the SPP propagates in the

direction, it will be damped

by Joule losses to the metal. This leads to the expression for

to be generally complex,

and the imaginary component of of

again provides the exponential decay of the SPP

as it travels along the interface. The imaginary part of k

is given by,

Im(k

) =

Im(ε

)

2 Re(ε

)



Re(ε

)

+ Re(ε

)



(2.17)

2. Background Theory

This gives the propagation length, L

as,

= λ

Re(ε

)

2π Im(ε

)



+ Re(ε

)

Re(ε

)



(2.18)

For a typical planar silver surface in air with

= 2

π/λ

= 2

π/

632

8nm, the propagation

length is

= 42

m. This propagation length is large enough so the SPP intensity is

suﬃcient to interact strongly and Bragg scatter on diﬀraction gratings with sub-micron

periodicity, a topic we shall cover in the next section.

2.4 Surface Plasmon Polaritons on Diﬀraction Gratings

2.4.1 Diﬀraction Gratings

A diﬀraction grating is an optical device in which the dielectric constant varies periodi-

cally across the surface, and whose period is of the order of the wavelength of light. This

periodic variation leads to the diﬀraction of light by way of localised phase changes in the

impinging ﬁeld, resulting in either the coherent constructive or destructive interference

of the waves in the far-ﬁeld.

The ﬁrst recorded observation of diﬀraction by ﬁne structure was by Francis Hopkin-

son in the late 1700s. Peering at a street lamp through a silk handkerchief, Hopkinson

observed rainbows and enquired with his friend, Rittenhause as to the cause. Ritten-

hause then went on to manufacture and study the ﬁrst reported ‘diﬀraction grating’

made of parallel hairs, set between two brass wires cut with a ﬁne screw head [

]. This

was the ﬁrst example of a transmission grating and the device’s dispersive nature.

In this thesis, a periodic modulation across a reﬂecting surface is achieved by surface-

relief. Shallow grooves are cut into a metal surface with a deﬁned period, on the order

of the wavelength of visible light, causing diﬀraction in the reﬂected light from the

surface. Other methods by which diﬀraction gratings may be produced exist, such as

phase-modulated diﬀraction gratings [

] or periodic hole arrays, but these will not be

dealt with here.

The coordinate system for a simple reﬂection grating of this type is shown in ﬁgure

2.5. The grating consists of a set of periodic grooves in a metal surface, with the

amplitude varying along the

-axis, with a periodicity of

. Light impinges on the

grating surface at a polar angle

, in a plane of incidence deﬁned at an azimuthal angle

, where

= 0

◦

corresponds to when the plane of incidence is perpendicular to the

grating grooves. The grooves are a depth

. The polarisation of the incident light

is deﬁned with respect to the plane of incidence, such that Transverse Electric (TE)

polarised light is when the electric ﬁeld,

T E

is oriented perpendicular (transverse) to

2. Background Theory

T M

T E

plane of incidence

Figure 2.5: The coordinate system used for a simple grating in the conical mount.

the plane of incidence, and Transverse Magnetic polarised light is deﬁned as the electric

vector E

T M

lying within the plane of incidence.

In this orientation, where

is allowed to vary and is not necessarily equal to 0

◦

, the

diﬀracted beams form in general a ‘cone’ of diﬀraction, and so it is named the ‘conical’

mounting/arrangement.

The relationship between incident and diﬀracted light in the conical mount is given

by the equation,

= (sin θ cos φ + sin θ

cos φ

) (2.19)

where

is an integer denoting the spectral order of diﬀraction,

is the incident

wavelength and

, φ

is the polar and azimuthal angle of the

diﬀracted order,

respectively.

The wavevector of light is deﬁned as

= 2

r/λ

, where

is the unit vector denoting

the direction of the wave. This wavevector is associated with the light’s momentum by

a factor of the reduced Planck’s constant, such that

~ k

, and it’s scalar equivalent

is the wavenumber (

). Similarly, the grating wavevector is deﬁned, in this case, as

= 2

x/λ

(its wavenumber is

). We may re-write equation 2.19 in terms of

2. Background Theory

these quantities,

−k

sin θ

cos φ

= k

sin θ cos φ − nk

(2.20)

= k

− nk

(2.21)

where

is the wavevector component of the diﬀracted light parallel to the surface and

is the incident light’s wavevector parallel to the surface. So a diﬀraction grating

modiﬁes the surface wavevector of the incident light by the addition or subtraction

(since n may be positive or negative) of an integer number of grating wavevectors.

Recall from section 2.3 that the coupling of free-space light to surface plasmon

polaritons is not possible on ﬂat surfaces due to the mismatch between the SPP and

light’s wavevectors (or, equivalently, the momentum mismatch). Since the diﬀraction

grating allows us to modify the wavevector of incident light, the diﬀraction grating may

be used to couple to these previously unmatchable modes. This will be explored in the

next subsection.

2.4.2 Coupling Surface Plasmons to Light by Coherent Scattering

The three illustrations in ﬁgure 2.6 show the momentum-space diagrams for monochro-

matic light of frequency

) incident on three gratings of diﬀerent pitch, at a

ﬁxed polar angle in the classical mount of

= 0

◦∗

. The allowed real momentum states

of propagating light are represented as a red circle of radius

. Conservation of energy

requires that any vectors representing propagating light have a magnitude of

, and

as such are represented as radial arrows of this circle. Points that lie outside of the

red light circle possess greater momentum than the incident light circle and represent

non-propagating (evanescent) light with imaginary momenta. The red stars on the

diagram are the allowed momentum states of a surface plasmon polariton. The surface

plasmon polariton momentum state lies at a ﬁnite real value of

, with no real value of

, indicating the surface mode has a purely imaginary value of

and so evanesently

decays normal to the surface while propagating along it. Notice the surface plasmon

polariton state lies outside the light circle, illustrating that even grazing light possesses

insuﬃcient momentum to match that of the surface mode.

The in-plane (

) component of light momentum on a grating surface is altered by the

addition of an integer number of grating vectors (equation 2.21). In ﬁgure 2.6(a), three

possible scattering events of incident light are shown, a +1

, a 1

and no scattering

). The allowed momentum states of light with these new values of

show that

∗

For the conical mount, these allowed momentum states would form instead a hemisphere in

three-dimensional k-space.

2. Background Theory

air

metal

SP P

+ k

− k

(a)

air

metal

SP P

+ k

− k

(b)

air

metal

SP P

+ k

− k

(c)

Figure 2.6: The allowed real momentum states for light incident on three diﬀraction

gratings. The gratings reduce in period (increase in

) from (a) to (c). Points along

the red circle are the possible momentum states for light in air (

). There is no

corressponding circle in the metal halfspace as light in this medium is considered to be

evanescently damped and so has no real value of momentum. The red stars show the

momentum states for SPPs.

2. Background Theory

diﬀracted light must leave the surface at a diﬀerent polar angle to the specular reﬂection

) to conserve total momentum,

(and so still lie on the radial line of constant

total momentum). This angle of diﬀraction is dependent on the size of the light circle,

determined by the frequency of the radiation, which illustrates the dispersive properties

of a diﬀraction grating. The grating represented in ﬁgure 2.6(b) has a periodicity less

than that of grating 2.6(a). The period has been chosen to illustrate the case where

the +1

diﬀracted order is grazing the surface. An increase in the polar angle, or a

decrease in the incident light wavelength would leave no available propagating light

momentum states for the +1

order. The order would become evanescent. To satisfy

energy conservation, this transition from propagating diﬀracted order to evanescent

light requires the power of that order to be re-distributed to the remaining propagating

orders. In spectra or angular data this is shown as a step in the intensity of reﬂected

light. First observed by Wood [

] and explained by Rayleigh [

], this feature is referred

to in literature as a diﬀraction edge, a critical edge or a Rayleigh anomaly. In this thesis

we will adopt the convention of calling this feature a diﬀraction edge.

Finally, ﬁgure 2.6(c) shows a grating of suﬃcient pitch where a +

scattering

photon has suﬃcient momentum to match the momentum state of a surface plasmon

polariton. This light’s value of in-plane momentum is

. The grating has coupled

free space light to the surface mode by enhancing the in-plane wavevector of the light

to match to that of the surface plasmon polariton. Momentum conservation is still

maintained as, while the real part of

= 0, the imaginary part is not. Using equation

2.19 this condition is expressed in this case as,

SP P

= k

+ k

(2.22)

Notice that this scattered light vector is now greater than the radial

circle, showing it

may no longer match a momentum state of propagating light and so does not propagate

away from the surface. The light is an evanescent diﬀracted order which resonantly

drives the surface plasmon polariton. This case may be represented on the dispersion

diagram for a SPP as shown in ﬁgure 2.7(a), where the zero-order SPP dispersion now

passes through the diﬀracted light cones (in blue). For all the points along the SPP

dispersion which lie inside the diﬀracted light cones (represented as blue circles along

the SPP dispersion (black line)), coupling can occur between the evanescent light and

the SPPs. Since all of this occurs in the non-radiative region, no optical eﬀect related

to the SPP excitation is observed in this case. However, this surface plasmon polariton

may also decay back into the zero-order reﬂectivity by itself diﬀracting by

−

. The

light returning to the specular order, having undergone two scattering events (and an

excitation and decay event), will be out of phase with the incident light leading to a

2. Background Theory

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0 k

0 2×10

6×10

10×10

(a)

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

−k

0 k

0 2×10

6×10

10×10

(b)

Figure 2.7: Two dispersion diagrams illustrating in-plane diﬀractive coupling to the (a)

zero-order SPP in the extended zone scheme and (b) the higher-order diﬀracted SPPs

couling to zero-order light in the reduced zone scheme.

sharp drop in the reﬂectivity observed. The observation of this interaction on the SPP

dispersion is shown in ﬁgure 2.7(b). Light diﬀracts and couples to the zero-order SPP,

then diﬀracts and decays back into propagating light inside the zero-order light cone.

For the SPP dispersion points lying inside the (red) light cone, we expect there to be an

eﬀect on light matching these momentum and energy values. It is exactly equivalent to

interpret ﬁgure 2.7(b) as the zero-order SPP curve (black) itself diﬀracting into the light

cone, as the surface waves themselves Bragg scatter to interact with zero-order light.

Typically this interpretation will be favoured in this thesis, as it allows a simpliﬁed

discussion for mode interaction without sacriﬁcing any scientiﬁc rigour.

This coherent scattering of SPPs will produce extrema in the specular reﬂectivity of

the grating indicative of the light having interacted with a SPP. By mapping the zero-

order reﬂectivity of a diﬀraction grating as a function of

and

, we may reproduce the

dispersion curves of the SPPs, with the reﬂectivity extrema occurring at the diﬀracted

SPP mode positions.

The shape of these dispersive bands is dependent on the plane of incidence, the

orientation of the grating and, in the case of bigratings, covered later, the additional

available scattering grating vectors. Figure 2.8 shows three possible intersections of

a plane of incidence with scattered SPP cones in (

ω, k

, k

) space and their possible

coupling. Since the SPP dispersion is approximated to be isotropic in all directions

∗

This is not generally true on grating surfaces, as the grooves have destroyed the isotropy of the

2. Background Theory

(a)

(b)

(c)

Figure 2.8: Intersections of the plane of incidence (

= 0) for (a) the unscattered

surface plasmon polariton cone (b) a surface plasmon polariton cone diﬀracted in the

plane of incidence and (c) a surface plasmon polariton cone diﬀracted perpendicular

to the plane of incidence. The projected intersection screen also shows the radiative

(white) and non-radiative (grey) regions of the light cone.

2. Background Theory

the SPP dispersion can be shown as a cone formed by sweeping the dispersion line

around the

axis. Figure 2.8(a) is the non-scattered zero-order SPP cone. The plane

of incidence and the conic-like intersection(along

in all cases here), is projected in a

back plane, with the region of propagating light shown in white, and the evanescent

light in grey. In this case, the experiment would yield the same dispersion as shown in

ﬁgure 2.3, with none of the available (blue) SPP momentum states passing through the

region of free-space light. In ﬁgure 2.8(b), a SPP cone has been scattered in the plane

of incidence by 1

. The projected intersection in the plane-of incidence shows that now

a portion of the SPP momentum states lies within the white region of free-space light,

and so observable coupling can occur. The shape of the SPP band is the same as the

in-plane scattered SPPs shown in ﬁgure 2.3.

Finally, ﬁgure 2.8(c) shows the results of a SPP cones having scattered in a direction

◦

to the plane of incidence. This can occur for a monograting oriented so that

= 90

◦

or this could be the result of the SPP undergoing a diﬀerent possible scattering event

on a square bigrating. The SPP momentum of this scattered mode does cross the region

of free space light (white area on the projected screen), and forms a ﬂatter band shaped

as shown which may couple to the free-space light. These ﬁgures 2.8(b) and 2.8(c) show

two possible functional forms of coupled SPPs in the plane of incidence. Real diﬀraction

gratings will contain many of these scattering mechanisms, and the mapped dispersion

will consequently be formed of multiple versions of these SPP bands.

2.4.3 Surface Plasmon Polaritons on Bigratings

Bigratings are deﬁned as diﬀraction gratings which possess two grating vectors which do

not lie collinear to each other. The grating equation from equation 2.21 then becomes,

)

= k

± n

(2.23)

where the variables are deﬁned as they were for equation 2.21, with

the grating

vector of the second pitch which is oriented at an angle

to the

-direction.

is the

integer number representing the spectral order in the

direction and

is the spectral

order in the

direction. In this thesis we shall occasionally use the notation (

, n

)

to refer to diﬀerent diﬀracted orders, for example a (+1

+1) scattered SPP would be

represented as k

SP P

= k

+ 1k

With this simple deﬁnition, all the possible 2D Bravais lattices for a diﬀraction

grating can be realised. With the angle

equal to the angle between the gratings in

real space (the corresponding angle in reciprocal space is

= 180

◦

− α

), we have the

following deﬁnitions listed in table 2.1.

surface on which the SPPs propagate.

2. Background Theory

Bravais Lattice α k

, k

References

Square α = 90

◦

| = |k

| [44, 52, 53, 76–78]

Rectangular α = 90

◦

| 6= |k

| [79–81][This Work]

Hexagonal α = 60

◦

| = |k

| [44, 58, 76, 82]

Oblique α 6= 90

◦

| 6= |k

| [This Work]

Rhombic α 6= 90

◦

| 6= |k

| [83]

Table 2.1: The Bravais lattice types and associated work on SPPs on such symmetries.

Rhombic is also called centred-rectangular, see Kittel [75] for an example.

Figure 2.9: The reciprocal space map for an oblique bigrating with grating vectors

and

. The black lines represent zero-order modes, the blue lines represent diﬀracted

modes. Lines represent grazing photon momentum states, dashed lines represent SPPs.

The coupling of light to SPPs on such a bigrating is best visualised in the

-space

diagrams for SPPs on such lattices. Figure 2.9 shows an oblique lattice in

-space with

= 60

◦

. The two grating vectors,

and

are illustrated as red arrows. The black

solid circle represents the momentum of a grazing zero-order photon and so the inside

of this circle contains all the possible momentum states of propagating free-space light.

The momentum states of this diﬀracted light are shown as blue circles, centred around

2. Background Theory

400 500 600 700 800

0.0 0.2 0.4 0.6 0.8 1.0

wavelength (nm)

T M

+1+2

diﬀraction edge

+1 SPP

inter-band transition absorptions

diﬀraction edge (weak)

+2 SPP

Figure 2.10: An example spectrum for a simple sinusoidal silver grating, calculated using

the Chandezon method. the grating has the following parameters:

= 750 nm, and

depth of 40 nm with a frequency dependent dielectric function for silver from literature

[

]. The spectra is taken at normal incidence with the polarisation perpendicular to

the grating grooves.

their respective lattice points and of equal size. The portions of blue circle which lie

within the zero-order black circle represent diﬀracted orders which may couple out and

travel away from the surface, the diﬀracted light. The dashed lines represent the SPP

momentum states. The black dashed SPP cone cannot be coupled to by free-space light,

while the diﬀracted SPPs (blue dashed) which lie inside the zero-order light circle may

be coupled with free-space radiation. These

-space diagrams also indicate the direction

of the group velocity of the surface modes, which travel away from their respective

lattice points and always lie perpendicular to the SPP contour.

2.4.4 Coupling Strength of Light to SPPs on Gratings

A typical specular reﬂection spectrum for a metal diﬀraction grating is shown in ﬁgure

2.10 and demonstrates some typical features observed in the reﬂectivity of a grating

supporting SPPs. This spectrum, produced using the Chandezon theoretical modelling

method outlined in chapter 3, is for a sinusoidal grating with a pitch of

= 750 nm,

and depth of 40 nm with a frequency dependent dielectric function for silver determined

from literature [

]. The spectrum is calculated at normal incidence with the polarisation

perpendicular to the grating grooves.

2. Background Theory

Features of interest in this ﬁgure are labelled on the graph. The diﬀraction edge,

the wavelength at which the ﬁrst diﬀracting order begins to (or ceases to) propagate

is clearly shown at 750 nm as a sharp critical edge. Slightly higher in wavelength, at

≈

765 nm, there is a sharp resonance in the reﬂectivity presenting as a minimum in

the reﬂected light. This minimum is a result of the reﬂected light being out-of-phase

with the light which is re-radiated from decaying SPPs. The incident light is scattered

by the grating and, when the coupling conditions are met, resonantly excites a SPP

which travels along the surface. This SPP may then decay and scatter back into reﬂected

light. A total of two scattering processes, an excitation and subsequent decay of a SPP,

give the total phase change of the re-radiated light of 90

◦

+ 90

◦

+ 90

◦

+ 90

◦

= 360

◦

relative to the incident light, while the reﬂected light which did not interact with the

SPP has accumulated a total phase change of 180

◦

by the simple reﬂection from a metal

surface. This results in the re-radiated light being out-of-phase with the reﬂected light

by 180

◦

, causing the observation of a minimum in reﬂection. Since no light is reﬂected,

the energy is lost to Joule heating.

The position of the second order diﬀraction edge is marked on the spectrum, but

is very weak. Beyond this diﬀraction edge, at

≈

420 nm there is a small resonance

exhibited as a region of enhanced reﬂectivity, a ‘bright spot’. This is the result of

reﬂected light being in-phase with the light re-radiated from the coupling of a 2

order SPP. The presentation of an optical interaction with a SPP as a maximum in the

reﬂectivity is again a result of the phase diﬀerence accumulated from the scattering,

coupling, and decay of light and the SPP. On a purely sinusoidal grating such as the

one modelled here, multiple scattering processes are required to couple to this SPP,

leading to the re-radiated light being in-phase with the non-interacting reﬂection. The

waves constructively interfere, resulting in a maximum in the reﬂectivity coeﬃcient.

The drop in the reﬂectivity at lower wavelengths is due to the metal’s natural

absorptions due to inter-band transitions and bulk plasmon excitations. The imaginary

component of the metal’s dielectric function at these wavelengths reﬂect these absorptions

by becoming much larger. This ‘background’ absorption can be seen in all the silver

grating results in this thesis.

2.4.5 Plasmonic Band Gaps

When a propagating SPP encounters a counter-propagating SPP with equal energy

and equal and opposite momentum, they will constructively interfere producing SPP

standing waves [

]. Typically, two possible SPP standing waves can occur, with

the nodes and anti-nodes of one SPP standing wave shifted

λ/

4 in space with respect to

the other, a typical result of any quantised standing waves conﬁned to a single value of

2. Background Theory

2λ

3λ

(a)

2λ

3λ

(b)

Figure 2.11: Cartoon of two possible SPP standing waves on a sinusoidal grating.

The blue and red regions show accumulation of positive and negative charge density,

respectively. These two standing waves, both which have a period of 2

, result in

diﬀerent ﬁeld arrangements and so are associated with diﬀerent energies.

momentum. This is analogous to the well known physical examples of electron standing

waves formed at Brillouin Zone (BZ) boundaries, or the vibrational modes of a solid

rod.

On a ﬂat surface, these two surface standing waves of electron density and coupled

electromagnetic ﬁeld are indistinguishable, save for the phase diﬀerence between them.

They occur at the same energy and momentum as one another. On a diﬀraction grating

the situation is quite diﬀerent. The arrangements of charge along the shaped surface

can result in the electron surface charge density sitting in very diﬀerent electromagnetic

potentials, for example at the bottom of a grating groove or at the top, which is

depicted in ﬁgure 2.11. In this case, the two possible standing waves have the same

period (enforced by the periodic lattice) and so the same momentum, but will be

associated with a diﬀerent electromagnetic energy. Between these two energy values,

SPP propagation is forbidden, due to destructive interference of the surface waves. An

energy region of forbidden SPP propagation has been opened up, a plasmonic band-gap.

This is shown in the extended zone scheme dispersion plot in ﬁgure 2.12(a) which shows

the Bragg scattering of SPPs at the ﬁrst BZ boundary (at

2) causing the formation

of a band gap.

The band gap shown in ﬁgure 2.12(a) will not be observed, as no available scattering

event can diﬀract this region into the zero-order light cone. However, the band gap

which occurs at the 2

BZ boundary may be, as shown in the repeated zone scheme

dispersion in ﬁgure 2.12(b)

∗

. In this case, while a single 1

scattering event may bring

∗

In this ﬁgure, the previous band gap at k

= k

/2 has been omitted for clarity.

2. Background Theory

angular frequency, ω

in-plane wavevector, k

−k

(a)

angular frequency, ω

in-plane wavevector, k

−k

(b)

Figure 2.12: The dispersion and band gap of (a) a SPP on a grating with one Fourier

Harmonic and (b) on a grating with the ﬁrst two Fourier Harmonics. The red line is

the light line for free space light, the blue curves show the SPP contours. In (a) the

black dotted line shown the unperturbed SPP contour, and the green dotted lines show

the position of the ﬁrst BZ.

the band gap region into the radiative zone, the interaction which leads to band gap

formation is between a zero-order and the 2

order SPP at the 2

BZ boundary

(

). This means that the required scattering of the SPPs must be a 2

event for

band gap formation. On purely sinusoidal gratings, which possess no 2

component in

their grating proﬁle, this scattering process is mediated by multiple scattering events,

which are very weak. Since this scattering amplitude for the SPP is so weak, the

interference between counter-propagating modes will also be weak, and band gaps are

unlikely to be large enough to be observed. If, however, the grating proﬁle contains a

component (by, for example, the addition of a secondary pitch of the grating equal

to half the period of the fundamental period), the SPPs may couple together to form a

standing wave via a single strong scattering event. It is the 2

component that couples

the SPPs together to form the band-gap, and the 1

component that scatters this into

the radiative region and so may be observed, as shown in ﬁgure 2.12(b).

Again, this is analogous to the formation of the electron band-structure of crystals,

formed when considering nearly-free electrons in a periodic potential [

], or photonic

band-gaps in dielectric stacks/photonic crystals [86].

A subtle diﬀerence in the deﬁnition used in this thesis, and common in the research

on surface plasmons, is that a ‘plasmonic band-gap’ might only occur over a limited

2. Background Theory

range of angles, whereas for electron band structure or photonic crystals a ‘band-gap’

occurs in all possible propagation directions, and the term ‘stop-band’ is used for

single-direction gaps. Here, we adopt the convention of a band-gap occurring if an

energy gap exists for which SPP propagation is forbidden over a given angle range, and

reserve the term ‘full plasmonic band-gap’ for forbidden SPP propagation for a range of

frequencies in all directions.

2.4.5.1 Coupling Light to Band Edges

To scatter SPPs in to the radiative region and also couple the two SPP modes together

to form standing waves requires suitable grating harmonics in the grating proﬁle. While

the magnitude and position of the band gap is determined solely by the presence and

size of these grating harmonics, the coupling of light to the SPP standing waves depends

strongly on the relative phase between the grating components. We can demonstrate

this simply by considering the band gap which will open at normal incidence (

= 0

◦

)

for a grating described by the simple Fourier sum,

f(x) = A

sin(k

x) + A

sin(2k

x + ψ

)

where, in our case,

= 5 nm,

= 2 nm,

= 2

π/λ

where

= 605 nm.

−

5 + 0

is the relative phase of the 2

component relative to the

component.

Three reﬂectivity plots as a function of angular frequency and in-plane momentum

around the band gap region are shown on the right side of ﬁgure 2.13, with each plot

showing the numerically calculated reﬂectivity for diﬀerent values of

. The method

used for calculation is the Chandezon method [

], detailed later in chapter 3. For

the ﬁrst case,

= 0

◦

, both band edges couple strongly to light, and are shown as a

minimum in the reﬂectivity, forming two symmetric blue bands. For

= +90

◦

, only

the upper frequency (higher energy) is coupled, while for

−

◦

, only the lower

frequency band edge is coupled.

The explanation for this phase dependent coupling is demonstrated in the ﬁgures

of the grating proﬁles and components, also shown in ﬁgure 2.13. The lower half

space of each of these sketches shows the relative phase between the two grating

components, and the upper half-space shows the resultant surface proﬁle by addition

of the

and 2

components. Firstly, by symmetry, the nodes and anti-nodes of a

SPP standing wave on such a grating will occur at either the peaks or the troughs of

the 2

component. Alternating nodes and anti-nodes at these points provide a SPP

standing wave which satisﬁes the symmetry of the grating, and also has a wave vector

equal to (

−

(

−k

))

2 =

at normal incidence (which is half way between the +

and

−k

scattered SPPs). One SPP standing wave charge arrangement will require

2. Background Theory

gratingcomponents

= 0

◦

in-plane momentum, k

−1

)

angular frequency, ω (rad s

−1

)

0.6

0.7

0.8

0.9

-1.5 -1 -0.5 0 0.5 1 1.5

2.98 3 3.02 3.04 3.06

× 10

(a) ψ

= 0

◦

gratingcomponents

= +90

◦

in-plane momentum, k

−1

)

angular frequency, ω (rad s

−1

)

0.4

0.5

0.6

0.7

0.8

0.9

-1.5 -1 -0.5 0 0.5 1 1.5

2.98 3 3.02 3.04 3.06

× 10

(b) ψ

= 90

◦

gratingcomponents

= −90

◦

in-plane momentum, k

−1

)

angular frequency, ω (rad s

−1

)

0.4

0.5

0.6

0.7

0.8

0.9

-1.5 -1 -0.5 0 0.5 1 1.5

2.98 3 3.02 3.04 3.06

× 10

= −90

◦

Figure 2.13: Sketches of diﬀerent grating proﬁles determined by the relative phases

of the

and 2

components (left), and the resulting coupling of the light to the

band-edges, shown as modelled reﬂectivity plots (right).

2. Background Theory

induced charge at the peaks of the 2

component, while the other solution will require

the charge to accumulate at the troughs.

In order for light to couple to the SPP standing waves, the electric ﬁeld vector

must have a component normal to the surface by which the impinging ﬁeld may induce

surface charge and hence couple to the surface electron density oscillations. In ﬁgure

2.13(a), the addition of the in-phase (

= 0

◦

) 2

component leads to points for both

the high energy (green dots) and low energy (blue dots) nodes and anti-nodes which lie

on the grating proﬁle where the gradient of the proﬁle is not zero. Because the grating

proﬁle at these points is not ﬂat, there is indeed a surface normal component which

can match to the impinging electric ﬁeld, and the result is that both the high energy

and low energy solutions may couple to incident light. In the second case in ﬁgure

2.13(b), where

= +90

◦

, the resulting surface proﬁle only has a non-zero gradient for

the high energy solution (green dots), while the position of the low-energy ‘hot-spots’

(the 2

minima) occur at a ﬂat region of the grating. Consequently, no coupling

occurs for the lower-energy mode, and only the upper band is coupled, as seen in the

reﬂectivity plot. Finally, for the

−

◦

case in ﬁgure 2.13(c), the resulting geometry

provides a normal component of the surface to the electric ﬁeld only at the blue dot

locations, corresponding to an induced charge arrangement for the low-energy band.

The high-energy arrangement points are located at the 2

component’s maxima, and

exist on ﬂat regions of the grating geometry. The coupling of light in this case is then

only to the lower-energy mode.

2.4.6 Polarisation Conversion

Linearly polarised light incident on a grating may be converted to the orthogonal

linear polarisation state in a process named ‘polarisation conversion’. Two mechanisms

have been identiﬁed for this conversion using diﬀraction gratings; one is mediated by

SPPs, and the other is due to the phase lag of the diﬀerent reﬂected polarisation state

reﬂections on deep gratings.

SPPs may mediate the polarisation conversion of incident light when the excited

SPPs travel along the grating surface in a direction of broken symmetry [

For the monograting case, this is any azimuthal angle other than

= 0

◦

or 90

◦

. In such

an orientation, both TM and TE polarised light provide a normal component of electric

ﬁeld to the grating surface and so may induce surface charge and resonantly drive

SPPs. Since this is the case, it is clear that SPPs decaying back into propagating light

may couple back out into either the TM or TE polarisation state, and so polarisation

conversion is possible. For shallow gratings, the maximum polarisation conversion occurs

at an azimuthal angle of

= 45

◦

, where the coupling to SPPs with TM and TE polarised

2. Background Theory

light is equal. For shallow gratings, the polarisation converted signal is proportional to

sin

), which is shown simply by the consideration of the the electric ﬁeld components

relative to the grating surface [

]. For bigratings, the same considerations hold true. A

SPP can mediate polarisation conversion when travelling along an axis with no mirror

symmetry, and cannot otherwise. For example, at

= 45

◦

on a square bigrating, no

polarisation conversion is expected.

The second method by which diﬀraction gratings may exhibit polarisation conversion

is again due to broken symmetry between the two orthogonal polarisation states [

This process is not mediated by SPPs and may occur on sub-wavelength, non-diﬀracting

gratings providing the grooves are suﬃciently deep. Consider the simple case of a

non-diﬀracting monograting, the electric ﬁeld incident on the grating surface may be

considered to decompose in two directions, one parallel to the grating grooves (

) and

one perpendicular (

⊥

) to the grooves. Since

does not cut across the grating grooves,

this ﬁeld will reﬂect as if it were reﬂected from a planar surface at a position equal to

the average plane of the grating. However, since

⊥

cuts across the grating grooves,

circulating ﬁelds may be produced in these grooves which alter the eﬀective average

plane of reﬂection for this ﬁeld component. The diﬀerence in position of these ‘eﬀective

mirrors’ produces a phase diﬀerence between the two components which serves to rotate

the plane of polarisation, leading to polarisation conversion. The maximum polarisation

conversion on such gratings occurs when the phase diﬀerence between the reﬂected

and

⊥

is 180

◦

, and so is a cyclic function with respect to the depth of the grating.

This necessity for the ﬁeld components to be 180

◦

out of phase requires the gratings to

be reasonably deep. For a gold sinusoidal grating illuminated with

= 632

8 nm with

a pitch of 300 nm the maximum polarisation converted signal occurs at a groove depth

of ≈ 180 nm [59].

Chapter 3

Theoretical Methods

3.1 Introduction

The calculation of the optical response of diﬀraction gratings has been an active pursuit of

electromagnetic research for over a century. In 1907, Lord Rayleigh proposed a treatment

for the reﬂected light from periodic surfaces by considering them a perturbation of the

case of a ﬂat surface [

]. This theory calculated that there must be a sharp discontinuity

in the reﬂectivity from the grating when a diﬀracted order of light emerges from, or

disappears beneath, the grating surface. This discontinuity, ﬁrst observed experimentally

as a dark band by Wood [

], corresponds to the instances at which power must be

redistributed to the propagating orders of light as a diﬀracted order either starts, or

ceases, to propagate at the grating surface. However, Rayleigh’s theory was inadequate

for all but the shallowest gratings, as it lacked a full consideration for the ﬁelds that

could occur in the grating grooves. Furthermore, Rayleigh’s treatment left the bright

band observed by Wood, characteristic of a SPP excitation, unexplained.

Since 1907, many methods by which to model the optical response of gratings have

been developed and have been veriﬁed experimentally. Methods such as Finite Diﬀerence

Time domain methods and Fourier modal-matching methods have all been tested. A

review of the these, and other, numerical methods of grating analysis can be found in

reference [36].

The purpose of this chapter is to provide a background to the methods used in

this thesis, namely the Finite Element Method (FEM) method and the coordinate

transformation method of Chandezon et al. [87].

3. Theoretical Methods

3.2 The Diﬀerential Method of Chandezon et al.

The method of Chandezon et al. [

] solves Maxwell’s equations by utilising a

coordinate transformation which maps the grating interfaces on to ﬂat, parallel, planes.

This avoids the Rayleigh theory’s failure to account fully for ﬁeld inside the grooves, as

the grating is eﬀectively ﬂattened by the coordinate transformation and the boundary

conditions are then matched across this ﬂat interface. For a monograting the transformed

coordinates are,

u = x (3.1)

w = y (3.2)

v = z − S(x) (3.3)

Where

(

) is the interface proﬁle of the grating. The ease of matching the boundary

conditions in this modiﬁed frame is achieved at a cost of the need for more complex

expressions for Maxwell’s equations in a non-orthogonal coordinate system of (

u, v, w

In this method, the derivation of the incident and scattered ﬁelds from Maxwell’s

equations are expressed as an inﬁnite series of plane waves, a Fourier expansion.

In order to solve this problem numerically, the inﬁnite Fourier series must be

truncated, placing a limit on the number of scattered ﬁeld components in the model. The

order of truncation is chosen from a compromise between obtaining suitable convergence

of the solution and minimising computation time. In this thesis, we primarily use

Chandezon method modelling to provide illustrative examples of reﬂectivity by which

we may explain certain observed optical features. For this purpose, we ﬁnd that limiting

the number of scattered ﬁelds in the system to 6 is suﬃcient.

The one-dimensional case was presented by Chandezon in 1980 [

]. Since then, it

has been extended to multi-layered gratings [

] and to bigratings [

]. The method

used in the thesis is the multi-layer bigrating method reported by Harris et al.[

]. This

allows for two crossed gratings of diﬀerent pitches to be considered, and the coordinate

transformation is altered from the equations 3.1, 3.2 and 3.3 to,

u = x (3.4)

w = y

= x cos α + y sin α (3.5)

v = z − S(x, y

) (3.6)

Where

is the angle between the two crossed gratings. This interface proﬁle

(

x, y

)

is represented by a Fourier sum. Because the technique is a diﬀerential method, sharp

discontinuities in the surface proﬁle gradient are diﬃcult to model. The number of

3. Theoretical Methods

Fourier components required to correctly simulate a step-like grating proﬁle can become

prohibitively large, although extensions to the method have shown than the method can

be applied well to trapezium shaped grooves [

]. For a precise description of a lamellar

grating as produced by our fabrication techniques a large number of Fourier harmonics

are required to represent

(

x, y

), which can make computation times prohibitively long.

However for the grating periodicities used in this thesis only the dominant scattering

associated with low order Fourier components need be considered to gain suﬃcient

insight into the underlying physical processes.

The Chandezon method has been used successfully to probe the electromagnetic

response of hexagonal bigratings [

], the eﬀect of groove shape on reﬂectivity resonances

due to SPP excitations [

], to examine asymmetric excitation of SPPs on blazed

gratings [

] and to determine the surface proﬁle of gratings from experimentally

acquired reﬂectivity measurements [43].

3.3 The Finite Element Method

The ﬁnite element method (FEM) for predicting the electromagnetic response of systems

is a numerical modelling process that subdivides a system into discrete elements, solving

Maxwell’s equations for each element and ensuring the solutions are consistent across

each elementary boundary [

]. Broadly, the process iteratively applies three steps

until a solution is found:

Determining the distribution and size of elements across the model. The continuous

array of elements is referred to as the ‘mesh’ of the model.

Numerically solving Maxwell’s equations for each element and for the mesh as a

whole.

The evaluation of the quality of the solution, to see if the model converges to a

consistent and physically accurate solution. If not, the process repeats from (1),

further reﬁning the mesh.

The software package used for FEM modelling in this thesis is the HFSS package

from Ansys, Inc. [

]. The software includes a computer aided design component,

which allows the building of three-dimensional models to represent the sample under

consideration. These 3D models may be assigned diﬀerent optical parameters, such as

frequency-dependent permittivities and permeabilities, and may be parametrised for

automated parametric alterations. The software’s default element shape is a tetrahedron,

with many tetrahedra forming a conformal mesh in the problem domain. In the case

of 3D models which contain curved surfaces, the software may also use curvilinear

3. Theoretical Methods

tetrahedral elements, greatly reducing the number of elements required to approximate

a curved surface.

3.3.1 Solving Maxwell’s Equations in the Mesh

The FEM package attempts to solve for the electric ﬁeld [

, for each element using

the wave equation,

∇ ×



∇ × E



− k

εE = 0 (3.7)

where

is the free space wavenumber and

and

are the relative permittivity and

relative permeability, respectively. As a numerical simulation, the software attempts

to minimise the left hand side of equation 3.7 to be as close to zero as possible for

each element, it will not in general, equal exactly zero. The magnetic ﬁeld,

is then

calculated using,

H =

ωµ

∇ × E (3.8)

where

is the angular frequency of the wave. The values for

and

must be found

for all the elements forming the conformal mesh in the problem space, and must be

consistent across each tetrahedron boundary. This leads to thousands of simultaneous

equations in the form of equation 3.7 for each ﬁnite element in the model, which are

combined into a matrix and solved for E numerically.

Once the ﬁelds have been determined, a generalised scattering matrix (S-Matrix) is

derived for the system, from which initial and ﬁnal optical properties such as reﬂectivity

or transmission can be extracted. The software may also attempt to ﬁnd the frequencies

at which the poles of the S-Matrix occur, and so calculate the eigenmodes of the system

and their associated ﬁelds.

3.3.2 Adaptive Meshing

The structure of the mesh used in the FEM model is critical to obtaining an accurate

simulation. In general, a higher resolution of mesh is required in areas where the electric

(and magnetic) ﬁeld vary rapidly. To achieve a suitable mesh, the FEM package uses a

process of adaptive meshing, reﬁning the mesh in areas where the error in the electric

ﬁeld calculation are high.

In the process, the software generates an initial, geometrically conformal mesh, and

solves equation 3.7 for the electric ﬁeld for each of the tetrahedra. Based on this initial

solution, the area of the problem domain where the exact solution has a large amount

of error is determined [

]. This is done by examining the residuals of equation 3.7,

3. Theoretical Methods

such that the approximate solution, E

approx

gives,

∇ ×



∇ × E

approx



− k

εE

approx

= residual (3.9)

for each tetrahedron. A set percentage of tetrahedra with the largest residuals are then

reﬁned by sub-dividing the element into even smaller tetrahedra. Another solution

is then found with this new mesh, and the process is repeated until the convergence

criteria has been met, or the total number of adaptive passes, set by the user to prevent

overly complex and memory intensive meshes, has been met.

It is important to note that the simple addition of elements to the problem is

ineﬃcient, and that the software’s determination of the reﬁnement areas by consideration

of highest tetrahedral residuals is not foolproof. Occasionally, the software may attempt

to reﬁne the mesh in locations that are not critical, but do lie in the upper percentile of

tetrahedral locations with high residuals. Users can make the simulation more eﬃcient

by deﬁning the initial mesh conditions using mesh operations in the software. For

example, someone setting up a simulation for plasmonic resonances in thin cavities may

start the initial mesh with a greater density of elements in the grooves (where they

expect rapidly varying ﬁelds) than the software alone might allocate.

3.3.3 Solution Evaluation and Convergence

The accuracy of the solution is determined by the model’s ‘convergence’. Because of

the direct relationship between the electric ﬁelds for a solution and the corresponding

S-Matrix, the convergence is measured by the change in the magnitude of two successive

solutions and is called the ∆

value. It is a measure of the change in electric ﬁeld

distribution between solutions. When the ∆

values becomes less than a user-speciﬁed

amount, the model is deemed to be ‘converged’ and the ﬁnal solution is obtained. An

example of mesh reﬁnement is shown in ﬁgure 3.1. Here, we ﬁnd that the reﬁnement

algorithm adds more elements to the mesh, to a total of over 40,000 in the ﬁnal case,

in order to reduce the ∆

value (red line) and achieve convergence. This particular

example is the convergence for a zigzag grating model, the results of which are detailed

in chapter 7. For the results in this thesis, the maximum ∆S value used was 2%.

3.3.4 Model Boundary Conditions

In the FEM package, the modelled system must be surrounded by appropriate boundary

conditions so a solution can be determined. Diﬀraction gratings in this thesis are

approximated to inﬁnite periodic structures, as the propagation lengths of SPPs is

far smaller than the sample size. Thus, a unit cell of the structure with appropriate

3. Theoretical Methods

2 4 6 8 10 12

number of adaptive passes

number of model tetrahedra

0 1 2 3 4

× 10

0 1.99 3.98 5.97 7.96

ΔS (%)

Figure 3.1: A graph showing the convergence of a typical FEM model as a function of

adaptive passes. The black line shows the total number of elements introduced in the

model, while the red line shows the ∆S parameter as a percentage.

boundary conditions is suitable for a full solution to the problem, an example being

shown in ﬁgure 3.2. This unit cell is surrounded by 4 types of boundary, labelled in the

ﬁgure. The ﬁrst type is called a master-slave boundary. This boundary pair constrains

the electric and magnetic ﬁeld so that ﬁelds at the master boundary must match exactly

the ﬁeld at the slave boundary. The periodic boundary conditions imposed by the

master-slave pair ensure that the solution is exactly equivalent to an inﬁnite array of unit

cells. The slave boundary may also replicate the ﬁelds at the master with an additional

relative phase delay. This phase delay is how the software calculates the ﬁelds for an

angle of incidence other than normal with the phase delay calculated internally from

the provided polar and azimuthal angles. The next type of boundary is the ‘Perfect

E’ boundary. This boundary acts as a perfect lossless conductor. It is placed on the

bottom of the metal ﬁlm, and acts as an easy way to double the thickness of the metal

ﬁlm by reﬂecting perfectly any impinging radiation. Since the ﬁeld penetration for

visible radiation in to a metal such a silver is very small (on the order of 20 nm), a

metal ﬁlm thickness of 50 nm, used together with the Perfect E boundary, provides a

suitable approximation to a semi-inﬁnite metal substrate without the additional cost of

increasing the number of tetrahedra. The ﬁnal boundary is a Floquet port. This acts

3. Theoretical Methods

Master/Slave 1 Master/Slave 2

Floquet port

Perfect E

Figure 3.2: Schematic of the diﬀerent boundaries used in the FEM model. The unit

cell shown is a zigzag grating (chapter 7) in red. The boundaries are highlighted and

named in each case. The Perfect E boundary has been extruded from the bottom of the

model for clarity.

as a radiation boundary that is open to free-space. The FEM package actually absorbs

any radiation that hits this boundary, without reﬂection, mimicking the character of

radiation escaping to free-space. This Floquet port also acts as an excitation source

and a way by which certain orders of diﬀracted ﬁeld are extracted, which is covered in

the next subsection.

3.3.5 Floquet Type Excitation

When extracting the reﬂectivity of a diﬀracting sample from the S-Matrix, it becomes

clear that simply examining the total ﬁeld arriving above the sample does not correspond

meaningfully to the types of experiments typically conducted in this thesis. This is

because the modelled ﬁelds would contain the zero-order reﬂection and all the diﬀracted

orders simultaneously, while experimentally we would only usually collect light from

the zero-order reﬂected beam. Floquet analysis allows us to extract the appropriate

order of diﬀracted light by recognising that the reﬂected light will be a superposition of

all these possible ﬁelds, whose in-plane momentum will be totally deﬁned by the polar

angle of incidence and the diﬀracting periodicities. The wavevectors of waves arriving

3. Theoretical Methods

above and at a plane parallel to the sample surface will be of the form,

= k

sin θ + nk

+ mk

(3.10)

Where

is the parallel wavevector,

is the free-space wavevector.

and

are the grating wavevectors in the

and

directions, deﬁned as

= 2

x/λ

and

= 2

v/λ

. Depending on the diﬀracted order, they have the integer multiples

and

, so, for example, the ﬁrst-order diﬀracted ﬁeld in the

direction would be

= 1

and m = 0.

Diﬀracted waves reﬂected from, or transmitted through, a sample must have well

deﬁned periods parallel to the surface and are determined by the diﬀracted orders, such

that,

n,m

) = E

n,m

+ nk

+ mk

) (3.11)

Where E

n,m

is the (n, m)

electric ﬁeld diﬀracted order.

The FEM package calculates all the possible propagating orders, based simply on

the periodicity of the surface and the wavelength of the incident ﬁeld, and is then able to

extract from the S-Matrix each ﬁeld solution corresponding to each order. Because the

periods of the ﬁelds in the simulation are now restricted, if there are strong eﬀects from

evanescent, non-propagating orders, (such as trapped SPPs) these must be included

as well. The calculation of which ‘Floquet modes’ to include is undertaken at a single

frequency, so it is important to calculate these using the highest frequency that is to be

simulated, to include all possible diﬀracted orders. The choice of the number of Floquet

modes to include is the equivalent of determining how many Fourier components of light

or of a surface to include for an accurate result, much like in the Chandezon case.

Chapter 4

Experimental Methodology

This chapter details the experimental methods used in this work for the fabrication and

measurement of metallic diﬀraction gratings supporting SPPs. Section 4.1 describes

the fabrication method for the gratings, a combination of electron beam lithography

(EBL) and template stripping. The second section explains the established methods by

which the reﬂectivity of gratings was measured as functions of both wavelength and

angle. Finally, section 4.3 introduces a new experimental method for the mapping of

SPP iso-frequency contours in momentum space, which was developed as part of this

work and has not been reported previously in the literature.

4.1 Sample Fabrication

The production of diﬀraction gratings used in this thesis has been achieved using a

combination of electron beam lithography and template stripping. Brieﬂy, this process

involves the exposure of resist to a diﬀraction grating pattern using a focussed electron

beam. This resist, when developed, is used as an etching mask for the silicon substrate,

producing a master template in a silicon ‘chip’. This chip is then metallised and the

pattern is transferred to a glass substrate by gluing the metal surface to the glass

substrate and removing the master with a razor blade. This leaves an inverse copy

of the master grating on the glass substrate, and the silicon master may be reused to

produce further copies.

An outline of the process is shown in ﬁgure 4.1. The details of the process are

explained in depth in the following sections.

4.1.1 Electron Beam Lithography

All electron beam lithography steps are performed in an ISO class 6 cleanroom. This

minimises the risk of small particle contamination and unwanted organic residues being

4. Experimental Methodology

electron beam

PMMA

Silicon Master

Reactive Ion Etch

3. Produced Master in Silicon

4. Thermal Evaporation of Silver

Glue

Glass Superstrate

Template Strip

Fabricated Grating

Glass Substrate

1. Electron Beam Lithography

2. Reactive Ion Etch

5. Superstrate Adhesion

6. Template Strip 7. Final Sample

Produced

Figure 4.1: Illustration of the fabrication method used for the production of diﬀraction

gratings.

4. Experimental Methodology

introduced to the samples.

A silicon wafer is prepared as a master substrate. Any large-particle dust is removed

from the wafer surface using pressurised nitrogen gas. The wafer surface is then spin-

coated in a

≈

600 nm thick protective layer of Poly(methyl methacrylate) (PMMA). The

typical protective polymer used was PMMA 950K A6, spun at 2000 RPM for 40 seconds

to produce the 600 nm layer. The wafer was then diced into 1 mm

chips which would

be used as the grating master substrates. The resist layer protects the chip surface from

the silicon dust that the dicing process produces. This layer, and the accompanying

contamination, is removed by inverting the chips in 100 ml of warm acetone for 10

minutes. After removing the protective resist, the chips are placed into a fresh solution

of boiling acetone (80

◦

C) for 1 hour, and then sonicated in 100 ml of isopropanol for 1

hour. The substrates may occasionally splinter during sonication, (introducing unwanted

Si dust to the solution), these chips are discarded. After 1 hour of sonication the chips

are removed from the isopropanol and dried using pressurised nitrogen gas. The clean

wafer chips are then inspected under dark-ﬁeld optical microscopy to ensure the surface

is optically clean. If not, the cleaning process is repeated.

The electron resist is then added to the surface. The resist used in all grating

fabrications included here is PMMA 950K A4. This is a positive electron-resist, whereby

exposure of the resist to the electron beam causes de-cross-linking of the polymer chain,

allowing the removal of the exposed resist with developer. The chip is placed in a

motorised spincoater, and 3-4 drops of PMMA are introduced to the surface using a

new, clean glass pipette. The chip plus resist is then spun at 4500 RPM for 40 seconds

to leave a 180 nm thick layer of resist on the surface [

]. The substrate is then baked

at 165

◦

C for 10 minutes on a hot-plate. This baking raises the resist temperature above

the glass transition temperature, and also causes the evaporation of any remaining

solvent (Anisol), improving substrate adhesion.

The substrate is then loaded into the electron beam lithography system and the

grating pattern is exposed. The system used is a nB4 Electron Beam Lithography

system (NanoBeam Limited [

100

]) and exposures are performed with a beam current of

2 nA

and a beam accelerating voltage of 80 keV. The exposure dose ranged between

400 − 600 µC cm

−2

, depending on the grating. Initially, the grating pattern was exposed

four times at diﬀerent locations on one substrate, each grating using a diﬀerent test dose.

The ﬁnal samples produced from these test areas are examined using scanning electron

microscopy and the grating’s electromagnetic response is measured using the optical

techniques listed in section 4.2, in order to determine the most suitable electron dose

to use for each particular grating. Once a suitable dose has been determined, future

gratings are produced with one grating per chip.

The electron beam write ﬁeld size, (the region exposed before the substrate needs to

4. Experimental Methodology

be moved) was

100 µm

, to minimise the eﬀect of height variation across the substrate.

This is the maximum write ﬁeld available in the system used. The nB4 electron beam

system has a theoretical beam diameter of

2.3 nm

at 100 keV, and the stitching error

between write-ﬁelds is on the order of 20 nm. This stitching error introduces an

extremely weak long-pitch periodicity to the samples, on the order of the write ﬁeld

size. The weak diﬀractive properties from these stitching errors are not found to inhibit

the optical eﬃciencies of the gratings signiﬁcantly, and are not observed experimentally.

The exposed sample then requires development to remove the de-cross-linked poly-

mer and provide a polymer etching mask for the Si. The developer used is a 15:5:1

solution of IPA:Methyl isobutyl ketone:Methyl ethyl ketone (IPA:MIBK:MEK) [

101

An endothermic reaction occurs when this solution is ﬁrst mixed, so to ensure devel-

opment times remain consistent between processes the developer must be allowed to

return to room temperature before use. The exposed chip is submerged in the solution

for 1 minute, followed immediately by a further 1 minute in a beaker of IPA. When

transferring the sample between the developer and the IPA, it is advisable to maintain a

small amount of the developer in a droplet on the sample surface, to prevent premature

exposure to the air. After the 1 minute in IPA, the sample is removed and blow-dried

with nitrogen gas. At this point the diﬀraction properties of the sample will be apparent,

and illumination with an appropriate wavelength laser allows an estimation/check of

the grating’s periodicity and orientation.

At this stage, the grating pattern has been transferred to the resist and successfully

developed. The sample consists of a coating of un-exposed PMMA layer, in which a

180 nm deep pattern has been developed. This polymer grating sits on a Si substrate.

The pattern is now transferred to the Si chip by means of reactive ion etching (RIE),

using the PMMA as a polymer mask. The reactive ion etching gas used is an oxygen,

CHF

mix, with the following parameters: 100 sccm

∗

CHF

, 3 sccm O

, at

38 mTorr

of pressure and 100 W power. This gives an etch rate into the Si of approximately

29 nm min

−1

The selectivity of Si to PMMA etching is suﬃcient for a 180 nm thick

layer of PMMA to be a suitable etching mask for depths of up to 80 nm etched into the

Si substrate. After RIE, any excess resist is removed by boiling the sample in acetone

and sonication in IPA. This leaves a master grating pattern in a Si substrate.

For ‘crossed’ bigratings, where a second periodicity runs along the grating surface at

an angle to the ﬁrst periodicity, this process is repeated with a previously fabricated Si

master grating used in place of the ﬂat Si substrate. This double exposure method was

preferred as the result is a grating where the two amplitudes have been summed, which

is consistent with the theoretical modelling undertaken with the method of Chandezon

∗

standard cubic centimetre per minute

4. Experimental Methodology

introduced in chapter 3.

4.1.2 Thermal Evaporation

To produce metallic diﬀraction gratings from the Si master, the master substrate is ﬁrst

metallised using thermally evaporated silver under high vacuum.

The clean template is placed in a vacuum chamber facing a molybdenum boat

containing 99.999% pure silver. The pressure in the vacuum chamber is reduced to

−6

mbar by use of both a rotary pump (down to 1

−2

mbar) and a water-

cooled turbo pump. At this low pressure, current is passed through the boat, which

heats, via Joule heating, to a temperature suﬃcient for the silver to evaporate, with

the silver vapour then condensing on to the structure. The rate at which the silver

evaporates is controlled by the current passing through the molybdenum boat. A typical

driving voltage of 70 V produces a deposition rate of approximately 3

−1

. The rate

of deposition and the nominal thickness is measured using a quartz crystal thickness

monitor.

The depth of the features for a grating produced by template stripping (described

in the next section) is pre-determined by the silicon master’s depth that was etched

via RIE. The amount of silver deposited on the master must then be greater than the

depth of the structures to ensure the sample is optically thick. It is found that for the

nominal depths of grating produced (ranging from 30 nm to 80 nm), a 300 nm thick

layer of silver was suﬃcient to ensure good pattern transfer and optically thick samples,

without producing unwanted silver ﬁlm tension that may cause the silver to ﬂake as the

layer relaxes during cooling.

4.1.3 Template Stripping

The method for transferring the master pattern to a glass substrate is based on the

method of Nagpal et al [

102

103

]. A glass substrate is prepared by ﬁrst swabbing the

surface with an acetone soaked cotton bud to remove any organic residue. The surface is

then drag-cleaned using lens tissue and IPA, removing any larger dust/cotton particles

before being examined under an optical microscope.

The metallised Si template is then positioned on a ﬂat, clean surface and a small

amount of UV optical adhesive (Norland 60 or 65) is introduced to the metallised

patterned area using a micro-pipette tip. Great care must be taken to ensure no

air-bubbles are present in this adhesive polymer, so it is advisable to de-gas the glue

ﬁrst by agitation in a UV free environment. The clean glass substrate is then placed

gently on the master and the glue is allowed to spread across the interface

∗

. Without

∗

If too much adhesive has been added, the glue may reach areas of the Si chip which are not

4. Experimental Methodology

moving the sample, the adhesive is cured under incoherent UV light provided by a

mercury lamp. The polymer is cured for a total of 20 minutes under this light

∗

After curing, the sample is turned over and a clean, sharp razor blade is inserted

between the glass (now the substrate) and Si chip (now the superstrate). With a small

amount of force, the master Si chip will be pried free and removed from the glass,

leaving behind the silver metal and, in it, the inverse metallic pattern. Once the Si

chip is freed from the glass/silver surface, the razor blade should be supplanted with

soft-tipped tweezers and the chip removed carefully, so as not to scratch or damage the

fragile silver surface.

Since silver degrades in the laboratory environment due to sulphur present in air,

this last step can be delayed until the sample is to be used. Until this point, the grating

pattern remains embedded in the Si/Silver interface, protected from the air and any

other possible contaminants.

Comparison between the master and sample SEMs shows less than 2% diﬀerence

between their mark-to-space ratios, indicating the high precision of copies taken from

the silicon master. The small diﬀerence may be attributed to the thermal relaxation of

the silver.

4.1.4 Polymer Replication

The embedding of gratings in a glass-like material can be desirable if the experimenter

wishes to access modes that may be lowered in energy in to the visible regime by use of

a higher refractive index. This is the case for a sample in chapter 7, and so the details

of embedding the grating in a high index material is covered brieﬂy here. The Si master

of the grating, prepared via EBL, and RIE, is coated with a UV curable polymer and a

thin, ﬂexible polymer sheet (acetate sheet). This combined polymer sheet and glue is

placed under pressure and cured rapidly using a high intensity UV source. The polymer

replica is then peeled oﬀ the master, and coated with a second UV curable polymer

(Norland 63). A clean glass slide is placed on top of this arrangement, and the glue is

cured, again by a UV light source. The polymer-master is peeled oﬀ the glass substrate,

leaving a polymer-replica on the glass substrate. This is then metallised via thermal

evaporation with a thickness of 300 nm of silver, and the bare side of the glass substrate

is adhered to a hemi-spherical prism using index matching ﬂuid. The resulting sample

is illuminated through the prism/index matching ﬂuid/glass/grating side, allowing for

metallised. In this case the template strip will be unsuccessful, and the process must be abandoned

at this point. The glass substrate is removed with great care and the master cleaned carefully by

submerging it in Acetone and then IPA.

∗

It is useful to include the pipette under the UV light also, as the residual glue present provides a

secondary check that the polymer is fully cured before attempting the template strip.

4. Experimental Methodology

the investigation of a grating sample embedded in a high index of n = 1.52.

4.2 Angle Scans

4.2.1 Monochromator

Pol

Sample

Pol

☼



Figure 4.2: System for the measurement of the reﬂectivity of a sample as a function

of both wavelength,

and polar angle

. White light is represented as blue lines,

pseudo-monochromatic light is shown as green lines.

To record the optical response of the diﬀraction gratings as a function of both incident

wavelength,

, and polar angle,

, a conventional rotational stage/monochromator

system is used. A schematic of the equipment is shown in ﬁgure 4.2. The incident

wavelength is selected by directing a white light source through the monochromator via

the focussing mirror, M1. Using an internal metallic diﬀraction grating and an output slit,

the monochromator outputs a beam of almost monchromatic light with a spectral width

25 nm, with a selectable wavelength range between

400 nm < λ

< 850 nm

This light is then collimated by a pair of mirrors, labelled M2 and M3 in ﬁgure 4.2, and

directed down the primary optical axis by a plane mirror located at M4. The light is

then spatially ﬁltered by a 0.5 mm pinhole (P) and passed through an optical chopper

3 k

. The chopper’s operating frequency is passed to

a pair of phase-sensitive detectors and is used as the triggering frequency for the signals

obtained from the detectors (D1 and D2) that are photomultiplier tubes. The beam is

then passed through a linear polariser (Pol) with which the polarisation dependence of

the optical response may be investigated.

For small-area samples, such as diﬀraction gratings fabricated using EBL, the grating

area may be as small as 2 mm

. In order to ensure the entire optical ﬁeld impinges

the grating, and not the surrounding material, the beam is focussed with a lens, L,

with a focal length of 0.5 m. The distance between the pinhole, P, and the lens, L,

is approximately 1 m, approximating to a collimated far-ﬁeld image with a radius of

4. Experimental Methodology

HeNe

C ND

Pol

LBS

Sample



Figure 4.3: System for the measurement of angular dependant reﬂectivity at a set

wavelength of λ

= 632.8 nm

0.5 mm, which is reproduced at the lens’s focal point. For smaller beam-spot areas, a

smaller pinhole may be used, at the cost of light intensity.

A small fraction of the light is directed to a reference detector, D1 via a plane-glass

beam splitter, BS. This reference signal allows for the normalization of the time-

dependence of the white-light source intensity. The transmitted beam continues along

the primary optical axis and hits the sample positioned precisely on a robotic rotational

stage, such that the sample position is central to the rotational axis and also at the

focal position of lens L. The robotic table, controlled via computer, sets the polar angle,

of the sample, with the spectral reﬂectivity detector, D2, mounted so that it moves a

corresponding angle of 2θ.

The spectrally reﬂected light is collected by the detector D2 after passing though a

second linear polariser. Both detectors, D1 and D2, are photomultipler tubes whose

output current varies linearly with incident light intensity. The signals from the detectors

are passed to a set of phase sensitive detectors which, together with the chopper triggering

signal, extract the reference and reﬂectivity measurement and pass the information to

the computerised recording software via an analogue interface card.

To obtain a reﬂectivity spectrum of a sample, the output voltages from both the

reference and reﬂection PSDs are recorded as a function of both angle and wavelength.

To normalise the resulting dataset, the sample is removed and the light directed into

the reﬂectivity detector at

= 0

◦

. The spectra of this ‘straight-through’ reference signal

is recorded, and then the original dataset is divided by the reference spectra, obtaining

the normalised reﬂection spectrum of the sample as a function of both wavelength and

polar angle.

4.2.2 Fixed Wavelength Scans

A similar set-up is used for acquiring angle dependent reﬂectivity of samples at a single

wavelength. In this set-up, shown in ﬁgure 4.3, a HeNe laser (

= 632

8 nm) is passed

through an optical chopper, C, a neutral density ﬁlter, ND, and a polariser before a

4. Experimental Methodology

Figure 4.4: Exploded view of the sample mount for use with a glass hemisphere. The

hemisphere is adhered to the glass substrate using index matching ﬂuid. The rig allows

azimuthal rotation of the sample and hemisphere.

beamsplitter (BS) directs a small fraction of the beam to a reference detector. The

beam transmitted through the beam splitter continues along the optical axis and is

passed through an optional lens before impinging on the sample placed, again, at the

center of rotation of a robotic table. The detectors used are photo-diodes, whose voltage

output is linearly dependent on the incident intensity. The signal from the detectors are

passed to a set of phase sensitive detectors, and the signal extracted, as before, using

the mechanical chopper as a triggering reference.

4.2.3 Embedded Samples

Some samples presented in this thesis are investigated with the grating embedded in a

high-index substrate. For such samples, a glass hemisphere arrangement is used (Figure

4.4). This allows illumination of the sample without the need for any corrections to the

polar angle which might otherwise be necessary because of refraction. The sample is

aﬃxed to the hemisphere using index-matching ﬂuid.

4.3 Iso-Frequency Contour Measurement Using Scatterom-

etry

A desirable measurement of the dispersion of SPPs on diﬀraction gratings is to record

the mode’s iso-frequency contours in reciprocal space. By mapping the position of

4. Experimental Methodology

0 k

−k

Figure 4.5: Three iso-frequency contours for the intersection of two SPP cones at angular

frequencies

, ω

and

. The contours show how the band-gap presents in

-space for

the cases of a frequency lower than the band-gap (

), inside the band-gap (

) and

above the band-gap (ω

the SPP contours in

-space, information is obtained about the 2D plasmonic crystal

lattice and the interaction of scattered SPP modes on the surface. These iso-frequency

contours are obtained as ‘slices’ through the 3D dispersion plot (

ω, k

) of the SPPs on a

grating for a constant frequency, and so are often referred to as equi-energy contours.

Examples of such iso-frequency contours are shown in ﬁgure 4.5. The upper ﬁgure shows

the intersection of two SPP dispersion cones (black lines), one scattered by +

and

the other by

−k

, plotted in the

ω − k

plane. At

= 0 these two SPP dispersion

curves meet and form a band gap, as shown in the ﬁgure. The dotted black line shows

how the SPP dispersion would have continued with no interaction. Three slices of

iso-frequency are shown,

, ω

and

, each chosen to illustrate a diﬀerent SPP contour

4. Experimental Methodology

type. The corresponding SPP contours in k−space are shown in the lower ﬁgure. The

frequency

is below the plasmonic band-gap in the dispersion, and the SPP cones

trace two distinct, approximately circular, contours in

-space. For

, the energy

slice is taken in the band-gap and the resulting SPP contours have ‘merged’, with

no SPP solution along the

= 0 line around the intersection. Finally,

shows an

intersection of the dispersion cones above the band-gap. This results in two contours,

one a larger merged ‘peanut’-shaped contour and a smaller SPP contour based around

= k

= 0. These SPP contours are analogous to iso-frequency Fermi-surfaces found

in solid state physics. This simpliﬁed ﬁgure is not periodic as it only includes two

scattered SPP cones for clarity, but the features observed for each energy slice will be

similar in all cases of SPP contour measurement. These contours show the presence of

band-gaps, the direction of SPP propagation (since the group velocity lies normal to the

contour) and gives information about the underlying scattering lattice. The complete

three-dimensional band structure of SPPs on periodic systems may be mapped by

collating the iso-frequency contours for a range of frequencies. The ability to map these

iso-frequency slices in momentum space is hence a powerful tool in the understanding

of a plasmonic system.

The most common method to map these iso-frequency contours is the use of reﬂec-

tivity plots for a range of azimuthal angle,

. By collecting reﬂectivity as a function of

polar angle

for a range of

, the position of the SPP mode in reciprocal space may

be mapped by the transformation

(

θ, φ

)

→ R

(

sin θ, φ

)

≡ R

(

, k

). This method

has several drawbacks. The accuracy of determining a SPP mode position in θ can be

limited if the plane of incidence lies parallel to a SPP contour, as the reﬂectivity minima

associated with the SPP will be broad. The resolution of this method is also limited by

the step size in

, which, for experimental purposes, may have to be limited to prevent

the experiment taking a prohibitively long time (particularly with silver samples, which

degrade over time).

Several methods to measure the reciprocal space map for plasmonic systems directly

have been previously published. Reports of using crystal defects [

104

] to couple to

the SPP modes have revealed the general shape of the underlying lattice of photonic

crystals. Direct recording of the ‘photonic Fermi surfaces’ using a plasmon topography

technique [

105

], which images a sample in the Fourier plane of an optical microscope

have been reported, as well as direct imaging of the contours using scattering through

transmission gratings [

106

]. Direct observation of dispersion in a single plane of incidence

using multiple wavelengths simultaneously is also found in the literature [

107

–

109

], but

these methods are not related to mapping the full SPP contours in k-space but rather

obtaining the in-plane dispersion curve directly.

In this thesis, we present a new method for directly obtaining the

-space contours

4. Experimental Methodology

of SPP at single frequencies by adapting an imaging scatterometer used commonly

in Natural Photonics investigations. This new method records the SPP contours of a

reﬂection grating across the entire light cone to a high degree of accuracy, and the results

can be numerically compared to theoretical predictions. By collating the contours from

a range of frequencies, the entire 3D band structure of the SPPs may be experimentally

determined.

4.3.1 Scatterometry

Scatterometry has been used previously to record the scattering patterns from samples

found in Natural Photonics, such as the naturally diﬀuse scattering of the Chrysochroa

fulgidissima beetle [

110

] or the Morpho aega butterﬂy [

111

]. It has also been used to

show the hexagonal shaped Brilloun zone of a pseudo-hexagonal lattice found in the E.

imperialis moth[112].

In this work, the previously reported experimental arrangement [

111

] has been

modiﬁed by the addition of spectral ﬁlters to limit the image acquired to a single

wavelength. Using a plasmonic grating sample, the scatterometer directly images the

iso-frequency surface of the plasmonic dispersion. By applying a simple deformation to

the image (also an adaptation new in this thesis) the scatterometer provides a map of

the coupled SPP iso-frequency contours in momentum-space over the entire incident

light cone.

4.3.2 Principle of Operation

A schematic of the scatterometry arrangement is shown in ﬁgure 4.6. White light

is directed through a collimating lens, L1 and an optional linear polariser allows the

investigation of the polarisation sensitivity of the acquired image. The white light is then

focussed through an alignment pinhole and the beam is reﬂected via the beam-splitter,

BS, on to an ellipsoidal mirror with an eccentricity of 0.833. The mirror, M1, focusses

the light onto the sample, positioned at G.

If aligned precisely, the cone of light focussed by the mirror will include light from

all directions (all azimuthal,

, angles) and a range of polar angle ranging from

θ ≈

◦

θ ≈

◦

. The lower limit of

is determined by the shadow cast by the sample itself.

The reﬂected light from the sample is then collected by the same ellipsoidal mirror,

M1 and is focussed through a second alignment pinhole, P2, positioned at the mirror’s

secondary focal point. This light is then collimated which causes the polar angle

to be

approximately linearly proportional to the obtained image’s radial axis

∗

. The beam is

∗

The small correction applied to the image that is required to obtain the true

(

θ, φ

) image is

detailed later in section 4.3.5

4. Experimental Methodology

Pol

L3 F L4

CCD

☼

Figure 4.6: Experimental arrangement for the modiﬁed imaging scatterometry system.

The angle

of the light impinging on the grating, G, is approximatly linearly proportional

to the image’s radial axis, r.

ﬁnally passed through a spectral ﬁlter and imaged using a CCD camera. The acquired

image is a directly mapped reﬂectivity plot of all polar and azimuthal angles,

(

θ, φ

with a high resolution determined by the pixel size and density of the CCD. Using a

range of spectral ﬁlters, the wavelength, and hence energy, of the map can be selected.

The ﬁlters used are: 450

5 nm, 500

5 nm, 550

5 nm, 580

5 nm, 600

5 nm,

650

5 nm, 700

5 nm and 750

5 nm. The reﬂectivity of the sample for a single

wavelength over the range 0

< φ <

◦

and 2

< θ <

◦

is thus recorded in a single

image by the CCD.

Light of appropriate in-plane momentum may couple to diﬀracted SPP contours

that exist within the recorded zero-order light circle. At the values of

and

which

match the momentum of such a SPP, a reﬂectivity anomaly will be found in the acquired

image. These reﬂectivity extrema, to a ﬁrst approximation, map the position of the

SPP iso-frequency contours over the large range of

and

available to this experiment.

4.3.3 Sample Preparation

Since the lower limit of polar angle,

is determined by the size of the sample area, a

small sample area is desirable to maximise the polar angle range. The grating samples

are prepared by template stripping from a Si master, (a process detailed in section 4.1)

to the centre of a large plate of glass, which acts as the substrate. The grating is then

further reduced in size by using a clean razor blade to remove excess sample area.

Most previously reported work using scatterometry use a glass pipette to place the

sample at the focal point of the ellipsoidal mirror. The pipette casts a characteristic

4. Experimental Methodology

Figure 4.7: Four raw images from the scatterometer for a wavelength of

= 650 nm.

The sample is a rectangular bi-grating supporting SPPs, as will be explored in chapter 5.

The polarisation states are (left to right) (1) polarisation

, which gives good contrast of

the thin dark SPP bands, (2) crossed polarisers with the ﬁrst polariser set to polarisation

and the second set at 90

◦

to this, named polarisation

(3) the alternative crossed

polarisation, with the ﬁrst polariser set to

and the second polariser set to polarisation

a, and (4) The ﬁnal polarisation, with both polarisers set to b.

shadow that may hide important spatial features. By using a large glass plate substrate

that covers the entirety of the the mirror face, shadows from alternative positioning

devices are avoided. It is found, by comparison of our ﬁnal images with calculated

diﬀraction edges and theoretical predictions, that any distortion eﬀects due to refraction

through the substrate are negligible.

4.3.4 The Role of Polarisation

The use of unpolarised light in this experiment is often adequate for the complete

mapping of SPP modes diﬀracted into the zero-order light circle. Greater contrast of

particular modes may be achieved by the use of two linear polarisers placed in the set-up

at the positions marked. The complicated polarisation arrangement, due to the focussing

from the ellipsoidal mirror, can cause ‘lobes’ of high and low reﬂectivity to appear, as the

multiple reﬂections on the mirror lead to polarisation conversion for all but two radial

axes. Still, if a single contour is of particular interest, the experimenter may adjust

the polarisers to obtain the greatest contrast between the reﬂectivity extrema and the

background. Depending on the polarisation states of the two polarisers, SPPs interacting

with light may present as a maximum, minimum or inﬂection of the reﬂectivity. This is

shown in an example dataset in ﬁgure 4.7.

These scattergrams are for a rectangular bigrating which will be discussed in chapter

4. Experimental Methodology

5. Notice that in the ﬁrst polarisation case (left most), rather ﬂat SPP contours are

seen clearly as dark contours crossing at the center (θ = 0

◦

), while for the second case

when the polarisers are crossed, the same SPPs exhibit as bright bands. This is a

demonstration of SPP mediated polarisation conversion[

]. Diﬀracted SPP contours

due to a second orthogonal pitch are best seen in the last polarisation case, and in this

case have a far smaller radius of curvature. Discussion of these particular experimental

results will continue in chapter 5.

The polarisation is chosen in each scattergram case depending on which SPP modes

are to be best observed, and this choice will be stated along with the results.

Notice in these scattergrams how 4 distinct dark lobes in the ﬁrst polarisation case

change to bright lobes in the second and third cases, and return to dark in the last

case. These are regions of polarisation conversion due to the mirror, and not due to the

sample.

4.3.5 Corrections and Momentum-Space Deformation

To obtain a map of momentum-space from the scattergrams, two image corrections are

required. The ﬁrst corrects the aberration of the ellipsoidal mirror to obtain an image

whose radial axis is linearly scaled with respect to the polar angle,

. This correction is

detailed in previous work [111].

The second deformation, scales the radial axis of the image to be proportional to

the in-plane momentum, such that the reﬂectivity plot, R(θ, φ) becomes,

R(θ, φ) → R(k

sin θ, φ) ≡ R(k

, k

)

Where

sin θ

is the in-plane momentum for a specular reﬂected photon in the plane

of incidence, at an azimuthal angle

. The eﬀect of this deformation is illustrated in

ﬁgure 4.8. This ﬁgure illustrates an important experimental consideration for the use

of scatterometry for the direct imaging of momentum space. Due to the

sin θ

radial

dependence of the ﬁnal

-space diagram, higher values of

provide less information

than the angles close to normal incidence. It is then possible to map a very large area

of momentum space, while only illuminating up to θ = 70

◦

− 80

◦

degrees.

4.3.6 Example Dataset

The processing of a raw dataset is shown in ﬁgure 4.9. The example shows a diﬀraction

grating with a period of 600 nm illuminated at a wavelength of

= 650 ± 5 nm

. In

ﬁgure 4.9(a), the raw image taken from the scatterometer is shown, with two dark

contours of reﬂectivity minima showing the position of the

±k

scattered SPP contours

4. Experimental Methodology

◦

0 90

◦

R(θ, φ)

(a)

◦

−k

0 k

−k

0 k

R(k

sin θ, φ)

(b)

Figure 4.8: The image correction for

(

θ, φ

)

→ R

(

sin θ, φ

). The black circles on both

diagrams show contours of equal angle, ranging from 10

◦

to 90

◦

in 10

◦

steps.

θ − φ

space. The image correction is then applied and the result is shown in ﬁgure

4.9(b), producing a momentum-space picture. Notice how the SPP contours have become

broader and towards the radial edge curve inwards to form a more circle-like dispersion

than they do in the raw image. Figure 4.9(c) shows the data adjusted for an azimuthal

angle of rotation so that the SPP contours lie in the (arbitrarily chosen)

direction.

This is to ensure our deﬁnitions between

and

remain consistent. Two blue lines

are added to ﬁgure 4.9(c), which are the calculated positions of the

±k

diﬀracted

light cones. The SPP contours, as expected, lie just outside these cones (see background

theory chapter 2). The contours sit closer to the diﬀraction lines along

= 0 than they

do at the extremes of the radius of the dataset, due to a small anisotropic dispersion of

the SPPs. This anisotropic dispersion is because this example diﬀraction grating is, in

fact, a complex grating type called a zigzag grating, which will be explained in detail in

chapters 7 and 8.

4. Experimental Methodology

(a) Raw data (b) deformation

Figure 4.9: The processing of a raw dataset from the scatterometer at a wavelength

= 650 nm. (a) Raw image obtained from scatterometer, (b) the image after the

applied momentum-space deformation, and (c) the ﬁnal image with corrected azimuthal

angle and addition of calculated diﬀraction circles.

Chapter 5

Optical Response of Metallic

Rectangular Bigratings

5.1 Introduction

Bigratings consist of two monogratings ‘crossed’ at some angle relative to each other [

The study of SPPs on such gratings has covered various symmetries, and various novel

optical eﬀects have been observed. Hole arrays that exhibit extraordinary enhanced

transmission that is mediated by SPPs travelling along the surface [

113

] are types

of bigrating, often with square symmetry. Full photonic band gaps for surface waves

has been demonstrated on hexagonal symmetry gratings [

114

], as has total absorption

of unpolarised light [77], and absorption of light across a broad angle range [115].

This chapter details the optical response of metal bigratings with rectangular lattice

symmetry. This symmetry of gratings has received comparatively less attention in the

published literature and serves as a good introduction to the methods and physics of SPP

on bigratings observed throughout this thesis. This chapter also shows experimentally

how anisotropic propagation of SPPs may be designed by controlling the scattering

amplitude in one direction and leaving the orthogonal direction unaltered.

The gratings used for this investigation, and all subsequent investigations in this

thesis, have straight-walled grooves and are commonly referred to as lamellar gratings.

It is instructive, then, to review the scattering properties of such lamellar type gratings,

and so a simple Fourier analysis giving the scattering amplitudes of a lamellar grating

is presented in section 5.3.1.

A fabricated rectangular bigrating is used to demonstrate the coupling to, and

observation of SPPs supported by such a structure. The reﬂectivity of a silver rectangular

bigrating is then used to experimentally map the SPP dispersion in section 5.3.2. The

observed coupling of light and the interaction of the SPP modes is explained with

5. Optical Response of Metallic Rectangular Bigratings

respect to the available scattering harmonics present in the constituent gratings. These

scattering harmonics are inferred from the Fourier expansion found in section 5.3.1 and

SEMs of the sample. These results serve to demonstrate the experimental techniques

used to map the SPP dispersion and also how the mark-to-space ratio is a key parameter

in the understanding of the scattering on such lamellar gratings.

The observation of polarisation conversion is also experimentally studied in section

5.4, showing that the reﬂected polarisation may be rotated when the plane of incidence

is not along an axis of high symmetry.

The new technique of imaging scatterometry is used in section 5.3.3 to map the iso-

frequency SPP contours in reciprocal space for the grating. The obtained iso-frequency

maps show the formation of a band-gap at the ﬁrst BZ boundary, a result that is

reproduced using a FEM model and shows good agreement. This model is used to

calculate the electric ﬁeld of the two diﬀerent SPP standing waves that occur at the BZ

boundary.

Finally, section 5.5 of this chapter shows how the deepening of one of the constituent

gratings provides a mechanism to control the anisotropic SPP propagation along the

surface. The deformation of the SPP iso-frequency contours is experimentally recorded

using imaging scatterometry.

5.2 The Rectangular Bigrating

A rectangular bigrating consists of two monogratings of diﬀerent pitches ‘crossed’ at an

angle

= 90

◦

. The coordinate system for this type of grating is shown in ﬁgure 5.1.

The plane of incidence is deﬁned at an azimuthal angle of

= 0

◦

when the wavevector of

incidence light, impinging at some polar angle,

, lies in the

plane. When the electric

ﬁeld vector of the impinging radiation is contained within the plane of incidence, the light

is said to be TM polarised, and when the electric vector lies orthogonal to the plane, it is

TE polarised. The

-direction is in the plane of diﬀraction for the longer-pitch grating,

which possesses a periodicity of

, and for all the gratings presented in this chapter is

equal to 600 nm. The grating vector for this grating is deﬁned as

= 2

x/λ

. The

second, shorter-pitch grating lies at an angle of

= 90

◦

to the

-direction, and has

= 400 nm in this chapter. As before, the grating vector of this short-pitch grating is

deﬁned as

= 2

y/λ

. The depths of the gratings are

and

, and are chosen in

this chapter so that d

= d

≈ 40 nm.

5. Optical Response of Metallic Rectangular Bigratings

T M

T E

plane of incidence

Figure 5.1: The coordinate system for a rectangular grating. Light is incident on the

surface at a polar angle

and an azimuthal angle

. The plane of incidence contains

the wavevector of the light, and the polarisations are deﬁned accordingly.

5.3 Dispersion of SPPs on Rectangular Bigratings

5.3.1 Scattering Components on Lamellar Gratings

In this chapter we use rectangular bigratings that have been fabricated using the electron

beam lithography and template stripping method outline in chapter 4. These have

surface relief grooves whose groove proﬁles are to a good approximation represented

by a rectangular step function. Gratings such as this are often referred to as ‘lamellar’

gratings, ‘binary’ gratings, or occasionally ‘rectangular gratings’. To avoid confusion

in this chapter, we shall reserve the label ‘rectangular’ for discussion of the bigrating

lattice, and refer to the groove proﬁle shape as simply ‘lamellar’.

To explain the SPP coupling and dispersion on rectangular bigratings we must gain

a qualitative understanding of the scattering strength from lamellar proﬁle gratings.

The strength of scattering for ﬁelds at the grating surface is determined by the Fourier

components of the surface proﬁle. This is because the incident ﬁeld’s wavevector will

be modiﬁed by the addition or subtraction of an integer number of surface proﬁle

wavevectors, and the diﬀraction eﬃciency into each order is proportional to the Fourier

coeﬃcients of these harmonics squared [

116

117

]. To fully understand the excitation

and interaction of SPPs on rectangular bigratings, we must ﬁrst obtain expressions for

5. Optical Response of Metallic Rectangular Bigratings

these Fourier harmonics present for lamellar gratings.

A step function representing a monograting surface proﬁle can be expressed by the

piecewise linear function,

f(x) =











−A, −L < x < −

A, −

< x <

−A,

< x < L

(5.1)

Where

is the amplitude of the grating,

= 2

is the period of the periodic function,

and

is a variable that aﬀects the groove widths. Two common ways of deﬁning the

relative size of the region of grooves to the peaks for such functions are the ‘mark-to-space

ratio’ (

MSR

) and the ‘duty cycle’ or ‘ﬁll-fraction’ (Γ) of the function. The MSR is

deﬁned as the ratio between the peak length and the trough length, while the duty

cycle is deﬁned as the fraction of the period which the peak occupies. The variable

the piecewise linear function was used for the convenience of calculation for the Fourier

series. MSR and Γ are deﬁned in terms of m as,

MSR =

2 − m

Γ =

MSR varies between 0 < M SR < ∞ while Γ varies between 0 < Γ < 1.

Several examples of the grating function (equation 5.1) are shown in ﬁgure 5.2(a).

This function is an even function and so may be expressed as a Fourier sum containing

only cosine terms, with coeﬃcients a

given by,

nπ

sin

nπm

(5.2)

The Fourier sum is then,

F (x) =

∞

n=1

nπ

sin

nπm

cos

nπx

The amplitudes of the ﬁrst four Fourier coeﬃcients as a function of the duty

cycle of the grating is shown in ﬁgure 5.2(b). The gratings fabricated in this chapter

were designed for a Γ = 0

5, which gives peaks equal in length to the troughs. For

Γ = 0

(

MSR

= 1), there are no even Fourier components present in the surface

expansion (

2,4,...

= 0). This means that, for a lamellar grating with Γ = 0

5, there is no

direct scattering of light by even scattering vectors, and the even-ordered diﬀraction is

5. Optical Response of Metallic Rectangular Bigratings

f(A, Γ)

-L 0 L

-A 0 A

(a)

0.0 0.5 1.0 1.5

Duty Cycle, Γ

0.00 0.25 0.50 0.75 1.00

n = 1

n = 2

n = 3

n = 4

(b)

Figure 5.2: (a) Three examples of unit cells for lamellar grating proﬁles represented

by equation 5.1.

(

Γ) for each case is

(

5),

75) and

25). (b)

Square of the Fourier coeﬃcients,

, deﬁned by equation 5.2, in lamellar groove proﬁles

as a function of Γ for

= 1

4. The two grey vertical lines highlight the two values

of Γ measured from the experimental sample used in section 5.3.2.

consequently very weak. Even-ordered diﬀraction in this case is not forbidden, however,

as light may scatter multiple times via odd-ordered scattering events (clearly present for

Γ = 0

5 in ﬁgure 5.2(b)), but the strength of this multiple scattering process is normally

quite weak. Small fabrication errors in Γ will lead to non-zero values of even Fourier

harmonics, and in this case direct scattering events will occur.

The expression of the Fourier components as a function of Γ is relevant to all the

gratings produced and measured in this thesis.

5.3.2 Experimental Mapping of SPP Dispersion

This section demonstrates the experimental mapping of SPP dispersion on rectangular

gratings using reﬂectivity measurements. An electron micrograph of the fabricated

grating is shown in ﬁgure 5.3.

This sample is a silver grating fabricated using electron beam lithography to produce

a silicon master, followed by the coating of the master with silver and then template

stripped on to a clean glass substrate. This process is detailed in depth in chapter 4.

The measured parameters from the SEM were

= 592

10 nm and

= 395

9 nm

with the duty cycle of the grating in the

direction measured as Γ

= 0

45 and in the

direction as Γ

= 0

54. The variation of Γ from the designed duty cycle of Γ = 0

will lead to possible direct scattering events for both odd and even grating harmonics,

5. Optical Response of Metallic Rectangular Bigratings

(a) (b)

Figure 5.3: (a) A scanning electron micrograph of the template-stripped rectangular

bigrating in silver. (b) A higher magniﬁcation of the surface showing small amounts of

surface roughness attributed to anistropic etching of the Si master. The parameters are

= 600 nm, λ

= 400 nm.

as can be seen by referring back to ﬁgure 5.2(b), in which these two values of Γ are

marked with grey lines.

The dispersion of SPPs on such a grating was experimentally mapped from the

sample’s zero-order reﬂectivity using the technique outlined in chapter 4. Using this

method, the re-radiated light from SPPs will in general diﬀer in phase to the zero-order

reﬂection and so produce reﬂectivity features that map the SPP dispersion. The polar

angle range used was 7

◦

< θ <

◦

in steps of 0

◦

, and the wavelength range used was

400 nm

< λ

800 nm in steps of 2 nm. The resultant reﬂectivity plots for diﬀerent

polarisations and azimuthal angles are shown in ﬁgure 5.4. Since both periodicities on

this bigrating will provide observable scattering at some point in our discussions, we

adopt the notation of using two integers (

m, n

) to represent the

scattering

events.

Figure 5.4(a) shows the measured reﬂectivity of TM polarised light for a rectangular

grating at

= 0

◦

. The blue lines show the momentum states of grazing photons for

various diﬀracted orders and in this ﬁgure include the

, +2

scattered and the

zero-order light line. No diﬀraction lines associated for the

period are present, as

the pitch of

was chosen to ensure, for the orientation of the grating at

= 0

◦

, there

was no diﬀraction present from the shorter period in this frequency range.

The ﬁgure shows dark bands of low reﬂectivity that map the positions of the

and +2

scattered SPP modes. These modes lie beyond their respective diﬀracted

5. Optical Response of Metallic Rectangular Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(-1,0)

(1,0)

(2,0)

(0,0)

(a) TM at φ = 0

◦

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(1,0)

(1,1)

(0,1)

(0,0)

(b) TE at φ = 90

◦

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(-1,0)

(1,0)

(2,0)

(0,0)

◦

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(1,0)

(1,1)

(0,1)

(0,0)

(d) TM at φ = 90

◦

Figure 5.4: The reﬂectivity for diﬀerent polarisations and azimuthal angles for a

rectangular grating mapped as a function of (

ω, k

) in the case of (a) TM polarised light

= 0

◦

, (b) TE polarised light at

= 90

◦

, (c) TE polarised light at

= 0

◦

and (d)

TM polarised light at φ = 90

◦

light lines, as the momentum of SPPs is greater than that of corresponding scattered

light. Several conclusions can be made about this grating from the observation of the

SPPs. Firstly, the

scattered SPP is coupled strongly to light, presenting as

a reﬂectivity minimum with only

≈

20% of the light being reﬂected. This is due to

the phase of the re-radiated light from the SPPs being out-of-phase with the direct

(non-interacting) reﬂected light from the surface. The +2

SPP is also coupled well

and is observed as a dark band. This suggests that there is a 2

component in the

grating surface proﬁle, as for light to interact with a +2

scattered SPP and present

as a dark band the light has undergone a single scattering process equal to +2

5. Optical Response of Metallic Rectangular Bigratings

which is only strongly observed if the proﬁle itself contains a 2

component. This is

expected for Γ

= 0

45, as measured from the SEM in ﬁgure 5.3. If Γ

= 0

5, there

would be no 2

component in the grating surface and the +2

could only present

in the reﬂectivity data via weak multiple scattering processes.

At the ﬁrst BZ boundary these two counter-propagating modes meet and form

a small band-gap, which is not fully mapped as it occurs at the upper limit of the

experiment’s frequency range. The observable part of this band-gap is the lower band

edge occurring at

π/λ

= 0

−1

. These SPPs coupling together to form a

band-gap means that the grating proﬁle also contains a 3

component which couples

these two modes together (see chapter 2). A 3

component in the surface proﬁle is

fully expected for lamellar gratings with Γ

= 0.45.

Figure 5.4(c) shows the same reﬂectivity map, this time for TE polarised light. TE

polarised light provides no normal component of electric ﬁeld to the surface of the long

pitch grating, and so no SPPs are excited. In the top corner of the dispersion, at larger

values of

and at high frequencies, there is a broad dark region that is attributed to a

largely over-coupled mode associated with the

scattered SPPs, for which the TE

polarisation is suitable for SPP excitation.

Rotating the grating by

= 90

◦

, so that the plane of incidence now lies along the

direction allows the observation of the

and

SPPs that present now as

out-of-plane modes when the light is TE polarised (Figure 5.4(b)). These mapped SPP

bands follow curved diﬀraction lines formed by the conic intersection of an out-of-plane

diﬀracted light cone (see the discussion in chapter 2).

A band-gap is observed in this direction between the (

0) and the (

1) scattered

SPPs, with the centre of the band-gap at

= 3

rad s

−1

. This corresponds to

an illumination wavelength of

= 546 nm, and we shall use a wavelength close to this

to observe this band-gap using scatterometry later in section 5.3.3.

These bands are coupled to well beneath these diﬀraction lines, and appear weakly

coupled once the band passes above the diﬀraction lines. This is due to the re-radiated

light from SPPs now being able to couple not just to the zero-order light, but also to

the allowed modes of diﬀracted light scattering along the

axis. This additional loss

mechanism of the light leaves less of the out-of-phase re-radiated light in the measured

specular reﬂection.

The straight lines in ﬁgure 5.4(b) show the diﬀraction lines associated with the

light and the zero-order light line. No SPP is observed that is associated with

the +1

light line, as the polarisation conditions are not met for the excitation of this

mode.

The ﬁnal dispersion map is shown in 5.4(d) for

= 90

◦

and TM polarised light. In this

case the SPPs associated with the

scattering events should be observed. Following

5. Optical Response of Metallic Rectangular Bigratings

(a) (b)

Figure 5.5: (a) A scattergram of a rectangular grating mapped to

-space for an

illumination wavelength of

= 550 nm. The red square indicates the position of the BZ

boundary. The white region bounded in the scattergram is replicated in (b), modelled

using the FEM.

the +1

light line, a dispersive mode, broader than the SPP bands seen previously,

is observed. This mode forms a large band-gap at

= 0, with a forbidden frequency

range observed between

3.2 × 10

rad s

−1

< ω < 3.5 × 10

rad s

−1

, suggesting a large

component in the grating which may directly couple the counter-propagating SPPs

together. Above this band-gap, a very large broad absorption band is seen, with the

reﬂectivity being reduced further inside the +1

diﬀraction cone. This is attributed

to largely over-coupled SPP modes, and the extra available loss channels inside the

diﬀraction cone. The small dark band observed at

ω ≈ 3.1 × 10

rad s

−1

is the lower

band edge of a band gap formed at the 1

BZ boundary in the k

direction.

5.3.3 Band Gap Observation Using Scatterometry

Scatterometry has been used in this section to observe the band-gaps that form at the

ﬁrst BZ in the

direction due to the interaction of a +1

and a

−

scattered

SPP. This is a new technique presented in this thesis, and the methodology is covered in

chapter 4. Figure 5.5(a) shows the results for a rectangular grating for an illuminating

wavelength of λ

= 550 nm.

In this ﬁgure, the BZ boundary is shown as a red rectangle and the

scattered

SPPs present as dark bands in the reﬂectivity. The

scattered SPPs are also

observed as extremely weak bright bands, due to the selected polarisation for this

scattergram, which was chosen to provide greatest contrast for the

scattered

5. Optical Response of Metallic Rectangular Bigratings

(−1, 0) (1, 0)

(a)

(−1, 0)

(1, 0)

(1, 1)

(−1, 1)

(b)

Figure 5.6: (a) Cartoon of the (

−

0) and (1

0) SPP iso-frequency contours at the

BZ boundary, from the original continuous SPP contours (dotted lines) to the split

modes (solid lines). (b) By considering the other interactions occurring at this point (or

by using the mirror symmetry of the BZ boundary), the continuous SPP contours are

recovered, leading to the observation of the SPP contour shape measured in ﬁgure 5.5.

SPPs. The

scattered SPPs are also observed as dark bands near the edges of

the plot at

= 0

−1

. This sample was produced from the same master as

the grating presented in section 5.3.2, so a 2

component in the surface proﬁle is not

unexpected.

As the scattered SPP contours cross at the BZ boundary, they are seen to split.

The contours intersect the BZ boundary perpendicularly, indicating that there is zero

group velocity in the

direction, as one would expect for a set of standing waves. The

shape of these contours is explained in ﬁgure 5.6 as the splitting of the modes due to a

band-gap.

The region of the band-gap is highlighted in white in ﬁgure 5.5(a), and in ﬁgure

5.5(b) the data in the area shown has been reproduced using a FEM model. The SPP

contour at point A appears discontinuous at this boundary, but this cannot be true. The

SPP contours are never discontinuous due to the translational symmetry of reciprocal

space as illustrated in ﬁgure 5.6, and this apparent discontinuity in the contour is in

fact the SPP simply being poorly coupled beyond the point of the BZ boundary. The

SPP contour at A will connect with the SPP eigenmode associated with the (

−

scattered SPP contour, and the coupling of light to this eigenmode is suﬃciently weak

as to cause this apparent discontinuity.

The model shows good agreement to the obtained scattergram with the parameters

as follows:

= 35 nm

, λ

= 600 nm

, λ

= 400 nm with Γ

= 0

45, Γ

= 0

and silver parameters from literature [

118

]. From this model, the electric ﬁeld proﬁles

5. Optical Response of Metallic Rectangular Bigratings

(a) High-energy solution (b) Low-energy solution

Figure 5.7: Colour plots showing the magnitude of electric ﬁeld at a set temporal phase

(colour scale) and electric vector direction (arrows) for the (a) high energy and (b) low

energy SPP standing waves occurring at the BZ boundary.

in the

plane were extracted and plotted in ﬁgure 5.7. These band gap solutions

correspond to the points A and B in ﬁgure 5.5(b), where the SPP contour intersects the

BZ boundary perpendicularly, indicating the SPP mode possesses zero group velocity

in the k

direction, indicative of a standing wave.

The electric ﬁeld plots show that both standing waves have a wavevector such that

2, as is expected at the ﬁrst BZ boundary. One mode extends further into the

air, and is the ‘photon-like’, high energy solution (ﬁgure 5.7(a)). This shows that charge

accumulates mostly on the grating peaks, while the charge is localised to the grating

grooves in the lower energy case of ﬁgure 5.7(b), in agreement with the analogous case

of simple sinousoidal gratings discussed in chapter 2.

5.4 Polarisation Conversion on Rectangular Symmetry

The conversion of the polarisation of light is of some importance in many plasmonic

applications. The two mechanisms of polarisation conversion outlined in chapter 2 detail

how the geometry of a diﬀraction grating and the presence of SPPs on a surface may

mediate signiﬁcant changes in the polarisation state of the reﬂected light. Both these

mediating processes are possible when the light impinging on a grating surface does so

along an axis of reduced symmetry. The basic principle of this is quite intuitive; one

5. Optical Response of Metallic Rectangular Bigratings

does not expect any physical response of the system to possess a diﬀerent symmetry

to that with which it started. So, when the electric ﬁeld of an incident light beam lies

in a plane of mirror symmetry shared by both the light and a surface, it is correctly

expected that the reﬂected light’s electric ﬁeld will occupy the same plane. This oﬀers

us an intuitive understanding for a typical diﬀraction grating with a single periodicity

reﬂecting polarised light. We do not expect the polarisation of the TE or TM polarised

light to change upon reﬂection when the electric ﬁeld lies either parallel or perpendicular

to the grating’s grooves as this is when the polarised electric (and magnetic) ﬁeld vectors

lie in the grating’s mirror symmetry planes (when

= 0

◦

= 90

◦

). However, if the

light ﬁeld vectors do not lie in the mirror symmetry planes of the diﬀraction grating,

say by rotating the grating azimuthally by an angle which is not

= 0

◦

= 90

◦

, the

mirror symmetry considerations place no constraints on the rotation of polarisation,

and polarisation conversion may be observed. In fact, maximum polarisation conversion

of the incident light (for a monograting) occurs half way between these two angles, at

φ = 45

◦

This is a general symmetry description of polarisation conversion, which gives us

an intuitive insight as to when we might expect the rotation of light’s polarisation

to change, or when it will not. On a square bigrating, with

, there is an

additional plane of mirror symmetry, at

= 45

◦

. At this angle, again, no polarisation

conversion is observed and the polarisation state of reﬂected light is unchanged [119].

In the rectangular bigrating case, the plane of incidence will only be collinear with

planes of mirror symmetry when

= 0

◦

180

◦

270

◦

. We expect to observe rotation

of the polarisation state of incident light for any azimuthal angle which is not

= 0

◦

= 90

◦

. To highlight the diﬀerences in the rectangular grating to the square grating,

we choose here an angle of

= 45

◦

to investigate the grating’s ability to rotate the

polarisation.

To record the polarisation conversion of light from TE to TM (and from TM to

TE), spectra were recorded in the range 450 nm

< λ

800 nm in steps of 2 nm.

The incident light was passed ﬁrst through a linear polariser set to either TE or TM,

and then directed on to the grating surface, which was oriented at

= 45

◦

. The

zero-order reﬂected light was then passed through a second linear polariser, which for

the purposes of detecting polarisation conversion was crossed at 90

◦

with respect to

the ﬁrst polariser. Normalisation of the polarisation converted signal was performed by

dividing the reﬂectivity spectra with a straight-through spectra of linearly polarised

light (both polarisers set to the same polarisation).

Figure 5.8 shows the results of this experiment. The upper half of the ﬁgure shows

the reﬂectivity for uncrossed (black) polarisers both set to the TE polarisation, and the

and crossed (red) polarisers set to TE and TM respectively. Resonant features exist

5. Optical Response of Metallic Rectangular Bigratings

0.0 0.4 0.8

450 500 550 600 650 700 750 800

0.5 0.7 0.9

rss45x[rss45y == angle]

T E:T E

+ R

T E:T M

wavelength, λ

(nm)

Figure 5.8: Spectra of polarisation conversion on the rectangular grating at

= 45

◦

. Top

panel: The reﬂection of TE polarised light,

T E:T E

(black) and polarisation conversion

T E:T M

(red). Lower panel: The addition of these two reﬂectance curves gives a more

precise position of the SPP modes.

in both spectra characteristic of SPP interaction with the light. For the polarisation

conversion signal (red line) we see a large broad region of polarisation conversion

between

500 nm < λ

< 600 nm

, which we attribute to polarisation conversion mediated

by largely over-coupled modes at high frequencies, evidence for which we observed

previously in section 5.3.2. The second resonant feature at

≈

690 nm presents as an

inﬂection of the polarisation converted signal and is in the same region as the reﬂectivity

minimum in the R

T E

case (black line).

This highlights an important point when dealing with the mapping of SPP position in

systems of low symmetry. Normally, the measurement of the SPP energy or momentum

is accomplished by measuring the resonant minima (or maxima) position in energy or

momentum space. However, when SPPs propagate along axes of broken symmetry, the

phase of the re-radiated light into a single polarisation state is insuﬃcient to determine

the exact SPP location, as additional loss channels are available (The SPP may couple

out to a diﬀerent polarisation state, and the phase of this light with respect to the

5. Optical Response of Metallic Rectangular Bigratings

incident ﬁeld may be diﬀerent). This is highlighted in the lower portion of ﬁgure 5.8,

where the two obtained reﬂectivities have been summed to obtain the reﬂectivity of

both TE:TE and TE:TM polarised reﬂections. The result shows reﬂectivity resonance

minima which better position the SPP in wavelength, as some of the additional phases

have been accounted for through interference by the summation of the two reﬂectivities.

The grey lines across both ﬁgures show that in the original upper spectra, the mode

positions do not correspond to this corrected (bottom) spectra.

5.5 Controlling SPP Anisotropy Using Rectangular Bi-

gratings

It was demonstrated in section 5.3.2 that SPP band-gaps form in both the

and

directions on a rectangular bigrating. The size of these band-gaps is proportional to the

diﬀraction eﬃciency required to couple two counter-propagating SPPs together to form

the required SPP standing waves. This diﬀraction eﬃciency into a particular diﬀracted

order is also proportional to the Fourier components squared, and an expression for

these was derived in section 5.3.1.

Examining equation 5.2, it is possible to increase the diﬀraction eﬃciencies of a

constituent grating by simply increasing the grating’s amplitude. Since on a lamellar

type grating, each Fourier component is proportional to this grating amplitude, the size

of all the possible SPP band-gaps may be controlled by simply deepening the grating.

On a rectangular bigrating there are two constituent lamellar gratings oriented

orthogonal to one another. SPP band-gaps in one direction may be controlled by

deepening the appropriate grating, leaving the band-gaps that form in the orthogonal

direction along the other constituent grating largely unaﬀected. This should allow the

design of SPP anisotropy, where the SPP dispersion varies largely depending on the

direction in which the SPP travels along the grating.

In this section, we deepen the short-pitch grating and observe the eﬀect of increased

SPP anisotropy using imaging scatterometery. The variable we adjust is

, and the

eﬀect on the surface proﬁle of increasing this is shown in ﬁgure 5.9.

Two bigratings were produced for this experiment, with the aim that they be

identical save for the depth of the shorter-pitch grating,

. The two gratings are

manufactured at the same time via electron beam lithography, using the same electron

dose to expose the pattern. The long pitch is

= 600 nm and for the short pitch

= 400 nm. The target depth of the long pitch for both gratings was

= 40 nm,

which is achieved by reactive ion etching the masters in the same etching exposure run.

The second short-pitch grating was then added to the master and the depth varied

5. Optical Response of Metallic Rectangular Bigratings

(a) d

= 40 nm

(b) d

= 80 nm

Figure 5.9: Diagram of the eﬀect on the surface proﬁle by increasing the depth

the rectangular bigrating from (a)

= 40 nm to (b)

= 80 nm. A larger value of

increases the diﬀraction eﬃciency of the constituent k

grating.

between samples. For the ‘shallow’ bigrating, the target depth of 40 nm was used, for

the ‘deep’ bigrating the target depth was d

= 80 nm

The expected eﬀect on the SPP iso-frequency contours is shown in ﬁgure 5.10. The

increase in the diﬀraction eﬃciency of the

±k

scattering events will be the dominant

eﬀect observed for the chosen grating’s parameters. By increasing the diﬀractive coupling

between SPPs, the (

0) SPPs will interact and form larger band-gaps at the 1

boundary in the

direction. This is shown in the ﬁgure as a relatively weakly coupled

SPP contour (yellow line) evolving into the stronger coupled SPP contour (red line).

The eﬀect of this is to ‘ﬂatten’ the band along the

axis. The interactions between

these contours and the (

0) are not included in this picture as they will require a

multiple scattering process by which to interact, and consequently are very weak. A

more detailed description as to the presentation of these bands in

-space is discussed

in chapter 4, under the experimental methodology for imaging scatterometry.

Figure 5.11 shows the experimentally mapped iso-frequency contours for the two

rectangular bigratings at a wavelength of

= 700 nm. Figure 5.11(a) shows two

SPP contours as a dark bands of reﬂectivity closely following the

scattered

diﬀraction cones (blue lines). The SPP contours in ﬁgure 5.11(a) exhibit a small degree

of anisotropy with respect to their dispersion, with the SPP contour closer to the

diﬀracted light lines at

= 0 than elsewhere along the contour. This is due to the

(

0) scattered SPPs interacting and forming band gaps with the (

, ±

1) or (

∓

, ±

SPPs. This interaction is strong as it only requires a single scattering event of

5. Optical Response of Metallic Rectangular Bigratings

(−1, −1) (+1, −1)

(+1, +1)(−1, +1)

(+1, 0)(−1, 0)

Figure 5.10: A sketch showing the expected iso-frequency contour deformation. Increased

coupling eﬃciency in to the

direction deforms the SPP iso-frequency contours;

represented as increasing in coupling eﬃciency from poorly coupled together (yellow

lines) to strongly coupled (red). The blue circles represent the diﬀracted light circles,

and the black circle represents the zero-order (un-scattered) light circle, which is the

area mapped by imaging scatterometry.

(a) d

= 40 nm (b) d

= 80 nm

Figure 5.11: Experimental iso-frequency contours for two rectangular bigratings at a

wavelength of

= 700 nm with (a) shallow orthogonal grooves of

= 30 nm and (b)

deep orthogonal grooves of

= 80 nm. The blue lines show the calculated position of

the diﬀracted light circles, which unperturbed SPP contours will follow.

5. Optical Response of Metallic Rectangular Bigratings

couple the SPPs together, which is a harmonic that is present in the grating’s surface

proﬁle. The strength of this interaction causes the deformation in the value for

SP P

for diﬀerent azimuthal angles. This interaction corresponds to the yellow SPP example

contour drawn in ﬁgure 5.10, as the (

0) modes which weakly interact with the

(±1, ±1) and (∓1, ±1) scattered modes.

By deepening the short-pitch grating to 80 nm this anisotropy is increased. This is

seen clearly in the experimental results in ﬁgure 5.11(b) where the (

0) scattered

SPPs interaction with the (

, ±

1) and (

∓

, ±

1) scattered SPPs have served to ﬂatten

the SPP contours, pulling them away from their associated diﬀraction circles.

In both cases the interaction between the (

0) and the (0

, ±

1) scattered SPPs is

not observed as the weak multiple scattering processes do not strongly couple the modes

together. Further, the (0

, ±

1) scattered SPPs are not seen in these scattergrams, as the

polarisation has been selected to highlight only the (

0) scattered modes for clarity.

Of interest is the eﬀect of this increasing band gap on the SPPs travelling solely

along the

direction. Since the deepening of the short-pitch grating aﬀects only the

diﬀraction eﬃciency in the

-direction, it is not expected that the SPP travelling solely

in the

-direction should be aﬀected. Experiments show, however, that this is not the

case. For the scattergrams in ﬁgure 5.11, along

= 0 we see that the SPP contour

lies further from the diﬀraction circles in the deep grating case when compared to the

shallow grating. Mapping the dispersion for these two gratings using the zero-order

reﬂectivity maps as a function of (

ω, k

) we obtain the dispersion of the SPP modes in

the plane of incidence containing k

, and these are shown in ﬁgure 5.12.

The SPP bands in the deep grating case are suppressed in frequency compared to

those obtained from the shallow grating. This frequency shift is seen more clearly in the

spectral plots and overlay of the dispersions shown in ﬁgures 5.12(c) and 5.12(d). For

the overlays of dispersion the reﬂectivity minima were taken as the SPP mode positions

(as it will be travelling along an axis of mirror symmetry, polarisation conversion does

not occur). These were extracted from the spectra of each plot for a given incidence

angle and collated together.

The results show that the eﬀective mode index of the SPP can be altered both in the

and

direction by increasing the depth of the short-pitch grating. This is analogous

to the work of Pendry [

] and the subsequent work on ‘spoof’ surface plasmons [

where surface structure is used to manipulate the asymptotic limit of surface wave

dispersion, introducing the concept of an ‘eﬀective surface plasma frequency’. We shall

return to this conclusion in chapter 7, where we demonstrate that the short-pitch can

be extremely sub-wavelength (

= 150 nm) and still aﬀect the propagation of SPPs

along grating surfaces.

5. Optical Response of Metallic Rectangular Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

1 × 10

2 × 10

3 × 10

4 × 10

5 × 10

2.5 × 10

3 × 10

3.5 × 10

4 × 10

-1

(a) d

= 40 nm

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

1 × 10

2 × 10

3 × 10

4 × 10

5 × 10

2.5 × 10

3 × 10

3.5 × 10

4 × 10

-1

(b) d

= 80 nm

0.0 0.2 0.4 0.6 0.8 1.0

angular frequency, ω (rad s

−1

)

2.4 2.6 2.8 3 3.2 3.4 3.6 3.8

× 10

(c)

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

1 2 3 4 5

× 10

2.5 3 3.5 4

× 10

40 nm

80 nm

-1

(d)

Figure 5.12: (a-b) Experimentally obtained dispersion diagrams mapped from the

reﬂectance of rectangular bi-gratings with nominal depths of (a) 40 nm, (b) 80 nm. (c)

The SPP mode position measured by reﬂection of TM polarised light at

= 7

◦

as a

function of angular frequency,

. The curves are for the shallow (black) and deep (red)

grating. The SPP mode shifts to lower frequencies as shown in (d).

5. Optical Response of Metallic Rectangular Bigratings

5.6 Conclusions

In this chapter we have introduced the concept of a bigrating as a grating with two

grating vectors which are not collinear. The example used in the chapter was the

rectangular bigrating, which possesses two grating vectors of diﬀerent magnitude,

oriented at α = 90

◦

The dispersion of these rectangular gratings was mapped from the reﬂectivity of

the samples in section 5.3.2. It was found that small errors in mark-to-space ratio

of such gratings lead to the strong coupling of even-ordered modes, which would not

be expected to couple strongly on such a surface if the mark-to-space ratio was 1. A

discussion as to the features associated with SPPs on such a surface has been presented.

This will be a background to future discussions on dispersion mapping throughout this

thesis.

Band gaps forming between the +1

and

−

scattered SPPs were experimen-

tally observed using imaging scatterometry, as presented in section 5.3.3, demonstrating

this technique for the ﬁrst time. FEM modelling of the gratings show the ﬁeld orientation

for these SPP standing waves which occur at the BZ boundary.

In section 5.4 it was found that such rectangular bigratings exhibit polarisation

conversion at

= 45

◦

, which is diﬀerent from conventional square bigratings. This is

attributed to the fact that SPPs propagating out of the plane of incidence inhabit an

axis of broken mirror symmetry, and so may decay into either polarisation state.

Finally section 5.5 investigates the eﬀect on SPP dispersion by deepening the short-

pitch of the rectangular bigrating. It is found that the anisotropy of the SPP mode can

be controlled by changing this depth, which controls the strength of coupling between

counter-propagating SPP modes. Finally, the mode index of the in-plane SPP (travelling

along the

direction) is also found to be aﬀected slightly by the deepening of the

orthogonal pitch.

Chapter 6

Optical Response of Metallic

Oblique Bigratings

6.1 Introduction

In this chapter, we examine the propagation of SPPs on periodic surfaces with the

lowest possible order of symmetry. The underlying Bravais lattice for these grating is

oblique, containing no reﬂection or rotational symmetry operations, save for the one

inversion operation shared by every possible Bravais lattice.

Section 6.2 discusses the oblique grating samples, their fabrication, the coordinate

system used and the reciprocal space lattice for an oblique grating. A comparison

between experiment and a theoretical model for a typical SPP iso-frequency contour

-space on such a grating is presented in section 6.3. The mapped dispersion of

SPPs on this geometry are recorded, with which their band-structure on the fabricated

oblique gratings are explained. Polarisation conversion mediated by SPPs is also found

on the oblique grating, with SPPs travelling in directions which are not in the plane of

incidence rotating the reﬂected light’s polarisation.

The most striking result in this chapter is that this low-symmetry class of grating

is shown to be the only exception for the formation of band-gaps at Brillouin Zone

(BZ) boundaries of all the 2D Bravais lattices. We show experimentally in section 6.5

that an SPP propagating across a BZ boundary does not split to form a band-gap, and

oﬀer a generalised symmetry discussion as to why this is so. The dispersion of SPPs

along axes containing the unique points of high-symmetry on an oblique grating are

obtained experimentally, and it is shown that for these discrete planes of incidence, SPP

band-gaps at BZ boundaries are still formed due to the symmetry of these axes.

6. Optical Response of Metallic Oblique Bigratings

T M

T E

plane of incidence

Figure 6.1: Coordinate system for an oblique bigrating. The typical design paramters

used in this chapter are λ

= 600 nm, λ

= 400 nm, α = 75

◦

, d

= d

= 40 nm.

6.2 The Oblique Grating

An oblique lattice is formed of an inﬁnite array of lattice points separated by two lattice

vectors of diﬀerent magnitudes, oriented at an angle with respect to each other

, such

that

α 6

= 90

◦

. To realise this symmetry using surface-relief gratings, two diﬀraction

gratings of diﬀerent pitches are ‘crossed’ at an angle

such that

α 6

= 90

◦

, forming an

oblique bigrating.

The coordinate system for this type of grating is shown in ﬁgure 6.1. The plane of

incidence is deﬁned at an azimuthal angle of

so that when

= 0

◦

the wavevector of

incidence light lies along the

-direction. When the electric ﬁeld vector of the impinging

radiation is contained within the plane of incidence, the light is said to be TM polarised,

and when the electric vector lies orthogonal to the plane, it is TE polarised. The

-direction is collinear with the grating vector

= 2

x/λ

for the longer-pitch

monograting, which possesses a periodicity of

. This period, for all the gratings

presented in this chapter, is

= 600 nm. The second, shorter-pitch grating lies at

an angle of

= 75

◦

along the

axis (deﬁned in the plane of the

plane at an angle

) and for the grating presented in this chapter has a period

= 400 nm. As

before, the grating vector of this short-pitch grating is deﬁned as

= 2

v/λ

. An

angle of

= 75

◦

was chosen to lie midway between the high symmetry cases of

= 60

◦

6. Optical Response of Metallic Oblique Bigratings

Figure 6.2: Example reciprocal lattices for an oblique grating.

= 600 nm,

= 400

nm and the angles between the two periodicities is α = 75

◦

(hexagonal-like) and

= 90

◦

(square/rectangular). The ﬁll fraction of the gratings is

designed as Γ

= Γ

= 0

5. The depths of the gratings are

and

, and are chosen in

this chapter so that d

= d

≈ 40 nm.

The reciprocal space map of the corresponding lattice is shown in ﬁgure 6.2 and

constitutes the lowest symmetry lattice set of all the two-dimensional Bravais lattices.

The reciprocal lattice of the oblique grating is itself oblique, with the reciprocal lattice

vectors deﬁned as

and

, oriented at an angle

= 180

◦

−

◦

= 105

◦

with respect

to each other. The only symmetry operation possible for this oblique lattice is a rotation

around a lattice point of 180

◦

, which for 2D lattices is the equivalent of a inversion

operation. Centred about each lattice point are circles representing various scattered

iso-frequency contours. The circles formed with solid lines show the contours for a

grazing photon, and the dashed lines represent the iso-frequency contours for scattered

SPPs which lie outside their respective grazing-photon lines (diﬀraction lines) due to

the greater momentum of SPPs compared to light. In this simple cartoon, the SPPs do

not interact and cross through each other unperturbed. However, if the SPPs interact

to form band-gaps these iso-frequency contours will deform as detailed in chapter 4.

The black circle is the case of the un-scattered zero-order light, which is the region of

-space accessible for mapping using the experimental method of imaging scatterometry.

The gratings for this chapter were fabricated using electron beam lithography (EBL)

6. Optical Response of Metallic Oblique Bigratings

Figure 6.3: SEM of an oblique grating master fabricated in a silicon wafer.

and the template stripping method outlined in chapter 4. A scanning electron mircograph

of a fabricated oblique grating master in silicon is shown in ﬁgure 6.3. The target param-

eters of this grating were

= 600 nm

= 400 nm, Γ

= Γ

= 0.5, d

= d

= 40 nm

6.3 Coupling of Light and SPP Mode Interaction on Oblique

Gratings

A square proﬁle grating such as those fabricated using EBL contain little or no even-

Fourier components in their surface proﬁle (see chapter 5). Consequently, the momentum

of a plane wave incident on such a grating may be modiﬁed with direct scattering events

with values of

±mk

, where

is odd. A direct scattering event, where the plane wave

is modiﬁed by

±nk

, where

is even, is not possible on a grating where Γ = 0

5. There

does exist, however, the opportunity for the plane wave to be modiﬁed by a total value

±nk

through multiple-scattering processes, summing the contributions from

±mk

scattering events. These multiple scattering events are found in experiment to be so

weak that they are rarely experimentally observed.

These scattering eﬃciencies also dictate the strength of interaction between counter-

propagating SPP modes. Modes that are separated by a direct scattering event will

interact strongly. Little, if any, interaction is observed for modes separated by the

weaker multiple-scattering process.

Figure 6.4 shows the theoretical and experimentally measured iso-frequency contours

for an oblique bigrating at

= 700 nm. Figure 6.4(a) shows the numerical prediction

using the Chandezon method, approximating the square groove proﬁles with the Fourier

6. Optical Response of Metallic Oblique Bigratings

(a) Model (b) Experiment

Figure 6.4: (a) The theoretically modelled iso-frequency surface for an oblique bigrating

illuminated with TM polarisation at

= 700 nm. Mode intersections labelled A-C

are discussed in section 4 (b) Experimentally obtained scattergram of the iso-frequency

contour imaged through crossed polarisers. The yellow circles overlay indicates the

theoretical mode position obtained from (a). Both colour bars range from low reﬂectivity

(black) to high.

sums,

F (x) =

n=1

nπ

cos

2nπx

(6.1)

F (y) =

n=1

nπ

cos

2nπv

(6.2)

Providing a suitable approximation to a lamella bi-grating with a depth of 2

= 40 nm,

= 600 nm,

= 400 nm,

= 75

◦

and Γ = 0

5. The dielectric function of silver

is taken from literature [

], and for the illuminating wavelength of 700 nm is equal

−

13 + 0

. The light in the theoretical plot is TM polarised for every

azimuthal angle. Only the

= 1

3 components are included in the calculation, as the

even components are considered to be absent for a grating with Γ = 0.5.

The corresponding experimentally obtained iso-frequency surface is shown in ﬁgure

6.4(b). The contour is obtained using imaging scatterometry detailed in chapter 4. For

the illuminating wavelength of 700 nm, the contrast of the entire SPP contour to the

background is weak, making the determination of the mode position in

-space diﬃcult.

6. Optical Response of Metallic Oblique Bigratings

To improve the contrast, the polarisers of the scatterometer are crossed, producing a

dark background reﬂectivity against which the polarisation conversion mediated by the

SPPs [

] provides greater contrast for the mode positions. The four bright lobes of

high reﬂectivity is polarisation conversion mediated by the ellipsoidal mirror in the

apparatus (see chapter 4).

The modelled values for the mode position (taken from the theoretical plot at the

position of reﬂectivity minima) found in ﬁgure 6.4(a) are plotted on the experimental

results in ﬁgure 6.4(b) as yellow circles, and show good agreement, especially considering

the simplicity of the model. This shows that the dominant scattering amplitudes in the

observable SPP band structure are only the n = 1, 3 components.

In the upper half-space of ﬁgure 6.4(a), three SPP contour crossings are labelled

A-C

∗

. Crossing point A is the meeting of a (

−

0) and a (0

1) Bragg scattered SPP

contour. These two SPP curves are separated in

-space by a minimum of two scattering

vectors, and so require a multiple scattering process to interact with each other. With

no 2

order harmonics in the grating proﬁle, the interaction is weak and no band-gap

is observed. The crossing point B is the intersection of the (0

1) and (1

1) SPPs. This

is a process by which the SPPs must scatter a total of 1

to interact, and so a small

band-gap forms. At point C, the (1

0) and (1

1) SPP cross. These are separated by a

single scattering vector 1k

, and so interact, forming a large band-gap.

The coupling eﬃciencies of plane-polarised light to all these SPP modes is strong

as the majority of the contours present are Bragg scattered back into the light circle

by a single scattering vector, which is inherent in the grating proﬁle. The exception

is the (1

1) SPP, (the contour between points B and C), which is scattered back in to

the light circle by a multiple scattering process. It is, however, observed quite strongly

between points B and C, though not strongly (and observed as a weak bright band)

outside of these points.

This coupling between points B and C is an example of SPPs ‘self-coupling’, the (0

and (1

0) SPPs coupling to the SPP contour close to the crossing points where there

is a component of the SPPs that fulﬁl the required energy and momentum matching

conditions for the (1

1) mode. These well coupled (0

1) and (1

0) SPPs resonantly

drive the (1, 1) SPP, resulting in its observation as a dark band.

These interactions can also be observed by mapping the dispersion relation of

the SPPs by recording the zero-order reﬂectivity from the sample as a function of

polar angle and wavelength. TE polarised light is incident on the sample and the

reﬂectivity measured for the wavelength range

400 nm < λ

< 850 nm

and the angle

range

◦

< θ < 75

◦

(see chapter 4). These experimental results are shown in ﬁgure

∗

These mode crossings are equivalent through an inversion symmetry operation to the crossings in

the lower half-space

6. Optical Response of Metallic Oblique Bigratings

Figure 6.5: The dispersion of modes on an oblique grating for

= 75

◦

, measured

using the reﬂectivity as a function of polar angle and wavelength. The mode posi-

tions (grayscale minima) and calculated diﬀraction edges (lines) have been coloured to

correspond to the scattered mode labels given in table 6.1.

(n, m) (-1,0) (1,0) (1,1) (-1,1) (0,1)

(-1,0) X

(1,0) X

(1,1) X X

(-1,1) X

(0,1) X

Light X X X

Table 6.1: The expected strong coupling (

) between SPP modes. The colour of

the (

n, m

) mode labels correspond to the highlighted modes in ﬁgure 6.5 scattered by

(

, mk

indicates coupling, but only for the orthogonal polarisation (TM) and

so is not observed in ﬁgure 6.5.

6. Optical Response of Metallic Oblique Bigratings

6.5 for an azimuthal angle of

= 75

◦

, with the plane of incidence containing

To facilitate the discussion of mode interactions, the dark-bands of low reﬂectivity

which map the SPP dispersion have been colour coded depending upon which dominant

scattering event the SPP contour is associated with. The associated diﬀraction lines

have also been coloured accordingly. These scattering interactions between the present

modes are summarised in the companion table 6.1.

The band gaps are clearly observed for the interactions between the (1

0) scattered

SPP band (green) and the (1

1) scattered SPP band (blue). Another large band-gap

is observed for the interaction of the (

−

1) scattered SPP (cyan) and the (

−

scattered SPP (red), which again are SPPs separated in

-space by a single grating

vector of 1k

A smaller band-gap is also observed for the (

−

0) and (1

1) SPP crossing (red and

blue bands, respectively). This coupling is mediated either by multiple scattering events

or a weak direct-scatter due to a small deviation in Γ away from the designed value

of 0.5, and consequently the frequency gap formed is small in comparison to the other

observed band-gaps.

The presence of a dark band associated with the (1

1) scattered SPP is clearly

observed in the mapped dispersion and coloured as a blue band of low reﬂectivity,

despite the (1

1) scattering event requiring multiple scattering with which to interact

with zero-order light. This region of the dispersion is observed for the same reason the

iso-frequency contour between points B and C in ﬁgure 6.4 is observed, a component

of the (

−

0) and (1

0) SPP ﬁelds present on the surface resonantly drive this mode.

The SPPs have ‘self-coupled’. A very small reﬂectivity minimum (not coloured) lies

between the red (

−

0) and the green (1

0) scattered SPP bands. There is also weak

self-coupling of these modes to the in-plane (0

1) SPP (its presence cannot be attributed

to the incident light ﬁeld, as the incident light’s polarisation state is incorrect for its

excitation for this orientation).

6.4 Polarisation Conversion

In this section we discuss how SPPs on an oblique bigrating mediate the rotation of the

incident ﬁeld’s polarisation. An example is experimentally demonstrated in ﬁgure 6.6.

This ﬁgure shows the reﬂectivity of an oblique grating as a function of (

ω, k

mapping the SPP dispersion by the characteristic minima in reﬂected intensity. For

subﬁgure 6.6(a), the incident polarisation is set to be TE polarised, and the detector’s

polariser is also set to TE. The corresponding polarisation conversion signal is recorded

by rotating the detector’s polariser by 90

◦

to TM, and is shown in ﬁgure 6.6(b). The

bright bands of polarisation converted signal lie along the same SPP contours, showing

6. Optical Response of Metallic Oblique Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(a) R

T E:T E

, φ = 75

◦

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.00

0.01

0.02

0.03

0.04

0.05

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(b) R

T E:T M

, φ = 75

◦

Figure 6.6: (a) Polarisation conservation and (b) conversion mediated by out-of-plane

scattered SPPs on an oblique grating.

that it is the surface waves which mediate this polarisation conversion.

The polarisation converted signal is normalised to the reference spectrum with both

polarisers set to TE polarised, and with this deﬁnition, only a maximum of

≈

5% of

the light coupled into the SPPs is re-radiated in to the orthogonal polarisation state.

Additionally, the SPP modes in the plane (which travel collinear to the wavevector

do not participate in this polarisation conversion as the SPPs are travelling along a

mirror-plane of the grating associated with the

direction (although there are no

mirror planes associated with the overall bigrating structure). This is an important point:

SPP mediated polarisation conversion occurs only when the SPPs are travelling along the

surface in a direction out of the plane of incidence and along a direction of broken mirror

symmetry. An oblique bigrating always provides out-of-plane scattering mechanisms

and so polarisation conversion will always be observed for out-of-plane scattered SPPs.

If, however, the plane of incidence contains a grating vector, conservation of momentum

dictates that the SPP associated with this grating vector will travel along in the plane of

incidence and polarisation conversion does not occur, even though the SPP is traversing

a surface of broken symmetry.

6.5 SPPs at the BZ Boundary of Oblique Bigratings

The formation of photonic band-gaps on an oblique bigrating is of interest, as in this

section we present results that show that on a surface with such broken symmetry

6. Optical Response of Metallic Oblique Bigratings

the locations in

-space of SPP standing waves do not necessarily occur at the BZ

boundary. This concept is known for the band-gaps that occur for electron propagation

in semiconductor crystals [

120

], but has not previously been demonstrated for SPPs on

gratings. To show this clearly, we must identify how photonic band-gaps are illustrated

in the iso-frequency contours recorded with imaging scatterometry. The iso-frequency

image obtained using scatterometry maps

-space at a single frequency, with SPP bands

seen as an anomaly in the reﬂected light. The group velocity of a general propagating

wave is deﬁned as,

= ∇

ω(k) (6.3)

where

is the group velocity,

(

) is the angular frequency of the wave as a function

of wavevector,

, and

∇

is the gradient operator with respect to

. For a small change

in frequency

dω

, the corresponding small movement in

-space,

, is related to this

group velocity simply by,

dω = ∇

ω(k) ·dk (6.4)

For an iso-frequency contour, there must be no change in frequency along the contour

(

dω

= 0). Setting

dω

= 0 restricts the values of

to those values that move along a

contour of equal frequency. It is then apparent that in the resultant expression,

·dk = 0 (6.5)

that

lies perpendicular to

. This is true for any general contour of constant

frequency. If the group velocity in one direction falls to zero at a boundary, such

as at the BZ boundary, the iso-frequency SPP contour will intersect that boundary

perpendicularly.

Plasmonic band-gaps on surface relief gratings are covered in chapter 2. To brieﬂy

recap, when a SPP meets an equivalent, counter-propagating SPP a standing wave forms.

There are generally two possible arrangements of the electric ﬁeld for a SPP standing

wave on a grating, which will generally diﬀer in energy. This leads to an upper and

lower energy band, with an energy range between where SPP propagation is forbidden.

The energy gap size is dependent on the energy of the two possible ﬁeld distributions,

and so is linked intimately to the surface geometry. The surface proﬁle also provides the

scattering mechanism by which SPPs Bragg scatter to meet counter-propagating SPPs

and form these standing waves. The strength of this scattering, and so the amplitude of

the Bragg scattered SPP, was discussed earlier in chapter 5 and aﬀects the magnitude

in energy of the band-gap.

Figure 6.7 shows the mapped iso-frequency contours of SPPs for the oblique grating

at a wavelength of

= 650 nm. The position of the SPP contours are found to present

6. Optical Response of Metallic Oblique Bigratings

Figure 6.7: An experimentally obtained iso-frequency contour map of SPPs on an oblique

bigrating with illuminated with incident light of wavelength

= 650 nm. SPP positions

are mapped as regions of low reﬂectivity. The blue circles indicate diﬀracted light circles,

and the green region is the typical BZ boundary drawn using the Wigner-Seitz method.

(a) (b)

Figure 6.8: Regions of interest from ﬁgure 6.7. (a) The (

−

0) scattered SPP contour

at the BZ boundary and (b) the (+1, 0) scattered SPP contour at the BZ boundary.

6. Optical Response of Metallic Oblique Bigratings

as bands of low reﬂectivity, with the polariser of the experiment chosen in this case to

best couple light to the (

0) modes. Also annotated on the ﬁgure are the diﬀracted

light circles (blue lines) and the BZ boundary formed using the Wigner-Seitz method

[75] (green line).

It is observed that the SPP contour passes through the BZ boundary seemingly

unperturbed. Figure 6.8 shows the two regions in which this occurs in greater detail.

In ﬁgure 6.8(a) the SPP contour following the (

−

0) diﬀracted light circle is shown

to pass through the BZ at an angle which is not perpendicular to the boundary. This

means that at the boundary, the SPP’s group velocity in the (0

1) direction is not zero,

and no standing-wave states in this direction have formed. The (0

1) scattered SPP is

not observed in this ﬁgure as the polarisation of the illuminating light has been chosen

to only couple strongly to the (1

0) SPP. Additionally the (

−

0) and (uncoupled)

1) SPPs are not seen to interact, separated as they are by a weak multiple scattering

process.

The same behaviour is observed in the second region in ﬁgure 6.8(b). Here, a

band-gap has formed inside the BZ

∗

, but the group velocity at the BZ boundary is still

ﬁnite. This highlights the fact that the mode crossing positions on an oblique grating

do not coincide with BZ boundary constructed using the Wigner-Seitz method.

A Brillioun Zone boundary in reciprocal space outlines a unit cell in the reciprocal

lattice and contains on the boundary points of high-symmetry. One way to determine a

boundary that contains the maximum amount of high-symmetry points is to connect

the perpendicular bisectors of the vectors connecting the nearest neighbours to one

lattice point, a method known as the Wigner-Seitz method. The area mapped out by

this method, for the highly-symmetric cases of square, rectangular or hexagonal lattices,

is exactly equivalent to the area we call the Brillioun Zone, and no other choice of unit

cell contains as many points of high-symmetry in the boundary.

In the oblique case, we may again use the Wigner-Seitz method to construct a

perfectly valid BZ, shown in ﬁgure 6.9 as the area bounded by a red line. In this ﬁgure,

it is seen that the only points of high symmetry on the oblique lattice are the singular

points lying midway along the perpendicular bisectors, and are labelled

, i

and

These points are equivalent to the points

∗

, i

∗

and

∗

by a rotation operation of 180

◦

and are isolated points of symmetry. However, this unit cell is not a unique choice, we

can easily contain these points of high-symmetry lying on the boundary of a unit cell

by using instead a trapezoidal unit cell, as shown by a blue line. In conclusion: The

most primitive unit cell of an oblique lattice is not unique.

The arbitrary choice of the unit cell which contains the high symmetry points, and

∗

Recall that the SPP eigenmodes are unaﬀected by the coupling of the light, and so although some

branches of the SPP contours are uncoupled to light in this scattergram, the band-gap still exists.

6. Optical Response of Metallic Oblique Bigratings

Figure 6.9: Two possible primitive unit cells for an oblique lattice. The Brillioun Zone

(Wigner-Seitz) cell (red) and a simple trapezium (blue) contain on their boundaries the

points of high symmetry labelled i

, i

that both choices is as useful as the conventional BZ in the oblique case, highlights

that the properties of physical phenomena on the lattice are determined, not by the BZ

boundary, but by the symmetry operations of the lattice.

Neumanns principle with respect to our system requires that the physical properties

of phenomena associated with the grating possess the same symmetry as the point

symmetry group of the grating [

121

]. While we could discuss speciﬁc propagation

properties of the surface modes with respect to its lattice, let us instead generalize

these concepts to some arbitrary vector ﬁeld and see what restrictions the symmetry

of the grating places upon it. Figure 6.10 shows an arbitrary vector,

lying on the

boundary of a rectangular or oblique unit cell. In the case of a rectangular grating,

the mirror and translational symmetry allows the deduction of the other shown vectors

through various reﬂections in the

and

planes or rotations about the

point.

These vectors sum to give a magnitude of zero in the direction perpendicular to the zone

boundary. Whether this vector ﬁeld represents the SPPs momentum, group velocity

or Poynting vector, the conclusion is the same: a standing wave forms perpendicular

to the BZ boundary. In the plasmonic case, a SPP has Bragg-scattered and interfered

with a counter-propagating SPP. The interference between these two SPPs, which by

symmetry can be equal in magnitude and opposite in direction, forms a standing wave.

The symmetry considerations we have outlined show us that this is possible, with no

restrictions on a mirrored SPP occupying the same position and, importantly, that

their vector properties will have equal magnitudes and so interfere completely. The two

possible ﬁeld arrangements for SPP standing waves on gratings lead to two solutions of

diﬀerent energies with a forbidden band of no propagating waves between the two: a

6. Optical Response of Metallic Oblique Bigratings

Figure 6.10: Applying the symmetry operations of the rectangular BZ (left), an arbitrary

vector,

(lying on the BZ boundary) corresponds to seven other vectors of known

magnitude and direction. The summation of these vectors leads to no perpendicular

component of the vector at the BZ boundary. In the oblique case (right), there is no

such condition, the

rotation operation only placing constraints on three additional

vectors.

plasmonic band gap. These are observed experimentally as discontinuities of the SPP

curves at the BZ boundary, as was seen in chaptrer 5.

Using the same approach, we apply the symmetry operations of the oblique lattice

to an arbitrary vector ﬁeld in ﬁgure 6.10. With no mirror symmetry, the oblique lattice

possesses only translational and a two-fold rotation symmetry operations (a two fold

rotation operation is equivalent in two dimensions to an inversion operation). As shown

in the ﬁgure, there are no special conditions on the vectors lying along the BZ boundary

formed using the Weigner-Seitz method, and no condition for the vectors to cancel

perfectly. Standing waves do not necessarily occur at the BZ boundary.

This leads to the observation that the band-gaps observed do not form at the BZ

boundaries, but simply where the SPP meets a Bragg scattered counter-propagating

SPP. The mid-points of BZ boundaries are of high symmetry in both rectangular and

oblique cases (points

, i

in ﬁgure 6.9), and do allow the total cancellation of our

vector ﬁeld. We can observe the points of zero group velocity at these BZ boundaries

by mapping the dispersion of the SPP mode in the special cases of

= 0

◦

and

= 75

◦

which place the plane of incidence intersecting points

and

in ﬁgure 6.9. In these

cases, we fully expect a band-gap to be present at the BZ boundary. These plots are

shown in ﬁgure 6.11. In the case of of a dispersion diagram such as this, zero group

velocity is shown as ∂ω/∂k

= 0.

6. Optical Response of Metallic Oblique Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

π λ

(a) R

T M

at φ = 0

◦

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

π λ

(b) R

T E

at φ = 75

◦

Figure 6.11: Dispersion plots mapped using the reﬂectivity of an oblique grating, showing

the occurrence of band-gaps at the Brillioun Zones along the special cases of (a)

= 0

◦

and (b) φ = 75

◦

6.6 Conclusions

In this chapter, SPPs propagating on an oblique bigrating have been investigated. The

dispersion of these surface modes has been mapped and the SPP interactions discussed

in terms of the available scattering amplitudes of the grating. Polarisation conversion

is observed on this grating, as SPPs travelling in a direction other than the plane of

incidence mediate the rotation of the light’s polarisation state.

Using imaging scatterometry, it is observed that the SPP contours are not perturbed

as they pass through the conventional BZ boundary. A generalized discussion on the

symmetry of the BZ is presented, concluding that this is because the BZ boundary on

an oblique grating is not a contour of high symmetry, and only contain isolated points

around which the symmetry conditions may be met for the formation of SPP standing

waves. Finally, when the plane of incidence intersects these unique points, SPP band

gaps may still be observed.

Chapter 7

Optical Response of Metallic

Zigzag Bigratings

7.1 Background

This chapter introduces a novel type of diﬀraction grating, the zigzag grating. A detailed

experimental and theoretical study of the SPPs excited on such a grating is explored,

with interesting results pertaining to the polarisation requirements of the coupled light,

the plasmonic band-gap character of interacting SPP modes, and the highly anisotropic

propagation of SPP on such a grating.

In 2005, Kleemann et al. [

122

] presented a new method for designing diﬀractive

optical elements with eﬃciencies and properties comparable to traditional surface relief

gratings. These gratings use sub-wavelength features perpendicular to the diﬀractive

periodicity to control the Fourier components available to scattering light [

123

]. The

symmetry and structure of these sub-wavelength features control the diﬀraction eﬃcien-

cies on such a grating, the way traditional surface relief gratings may ‘blaze’ their groove

proﬁle to achieve the same result [

]. These diﬀraction gratings can be produced using

well established lithography techniques to pattern the surface, a simpler task than pro-

ducing complicated and precise groove shapes on traditional blazed gratings. Kleemann

at al. named these gratings Area Coded Eﬀective medium structures (ACES), and the

versions exhibiting similar characteristics to blazed surface-relief gratings ‘BLACES’.

Surface Plasmon Polaritons were excited on such BLACES in 2009 by Bai et al.[

] and

showed that asymmetrical excitation of the SPPs could be achieved at normal incidence,

typical of the grating’s blazed character. It is established by this previous work that a

patterned metallic surface with sub-wavelength features perpendicular to the diﬀractive

periodicity is capable of manipulating the strength of diﬀracted orders and, consequently,

the coupling of SPPs to light. The use of the sub-wavelength structure to manipulate

7. Optical Response of Metallic Zigzag Bigratings

the optical response of these surfaces qualiﬁes them as a type of ‘metamaterial’ surface

[124].

In the work to date on BLACES, the propagation of the SPPs has been along

an axis of mirror symmetry. Other work has hinted at novel optical eﬀects, such as

‘magnetic mirrors’ [

125

126

] or spatial control of coherent anti-Stokes emission [

127

when plasmonic resonances are coupled to by light on geometries of broken symmetry.

In this chapter, we present a new ACES grating with sub-wavelength structure with

a diﬀracting periodicity along an axis of broken mirror-symmetry. We name this grating

a ‘zigzag’ grating. Coupling of light to SPPs on this grating is highly polarisation

selective and both TM and TE polarised light can excite SPPs propagating in the same

plane, which is theoretically investigated in section 7.3.1 and experimentally veriﬁed

in sections 7.3.3 and 7.3.4. Further, we ﬁnd that the symmetry of this grating places

restraints on the formation of plasmonic band-gaps at Brillouin Zone (BZ) boundaries,

which is explored in section 7.4 and that the propagation of SPPs is highly anisotropic

with respect to azimuthal angle (section 7.5). The degree of SPP anisotropy is so great,

that the zigzag grating is suggested as an excellent candidate for single-wavelength

collimation of SPP beams for use in plasmonic circuits. This highly anisotropic SPP

band structure leads to the observation of out-of-plane SPPs associated with the sub-

wavelength periodicity, which evolve to have cavity-resonance character for deep zigzag

gratings.

7.2 The Zigzag Grating

A diagram of a zigzag grating is shown in ﬁgure 7.1, including the coordinate system

used throughout this chapter. The zigzag grating is formed of a surface-relief grating

of sub-wavelength, and hence non-diﬀracting grooves that run along a silver surface.

Perpendicular to this non-diﬀracting grating, the grooves are perturbed in the surface

plane to introduce a long-pitch variation that may diﬀract visible light. This zigzag

perturbation introduces a diﬀracting pitch of

, which lies perpendicular to the short-

pitch of the surface-relief grating,

. In many ways, the zigzag geometry is similar

to a conventional rectangular bi-grating, however, the long-pitch is present due to a

zigzag surface perturbation (not surface-relief grooves). This makes the zigzag grating’s

optical response quite unlike that of a standard bigrating .

The plane of incidence is deﬁned at an azimuthal angle,

, which equals 0

◦

when the

plane is coincident with the plane of diﬀraction from the long pitch. The polar angle

of impinging radiation lies in the plane of incidence, and is deﬁned as

= 0

◦

when

the light is incident normal to the average plane of the surface. The polarisation of the

light is deﬁned as Transverse Magnetic (TM) polarised when the electric ﬁeld vector

7. Optical Response of Metallic Zigzag Bigratings

T M

T E

plane of incidence

Figure 7.1: The coordinate system of a zigzag grating. The experimental sample

parameters were

= 600 nm,

= 150 nm,

= 29

9 nm with the plane of incidence

deﬁned at an angle

and the polar angle of incidence as

. The two polarisation

orientations for the electric ﬁeld vector are also shown for TM and TE polarisations.

lies in this plane of incidence, and Transverse Electric (TE) polarised when the electric

ﬁeld vector lies orthogonal to it. The zigzag grating under consideration in this chapter

possesses mirror symmetry along the yz planes deﬁned at x = 0, λ

/2, λ

7.3 The Coupling of Plane Polarised Light to SPPs on

Zigzag Gratings

7.3.1 Theory

To examine how the orientation of the electric ﬁeld vector of impinging radiation may

relate to the excitation of SPPs on a zigzag grating, we consider the magnitude of the

electric vector that lies normal to the zigzag surface for the two polarisation cases of TM

and TE polarised light. The induced local surface charge is, to a good approximation,

proportional to the electric vector normal to the surface [

128

], so by considering how the

local normal of electric ﬁeld varies along the grating, we may infer what local surface

charge arrangements are possible on such a grating. At normal incidence (

= 0), the

wavevector of these diﬀerent charge distributions are then equal to the Bragg vectors

7. Optical Response of Metallic Zigzag Bigratings

(a)

ˆn

for

polarised light

(b) E

ˆn

for

y polarised light

Figure 7.2: The surface normal components of electric ﬁeld vector for (a) TM and (b)

TE polarised light (arrows) projected on a contour of the zigzag surface proﬁle. The

contour plot amplitude ranges from 1 (white) to -1 (blue).

required of the zeroth order SPP to scatter and interact with light.

A simple expression for the local normal component of electric ﬁeld can be obtained

by considering an approximation to the zigzag surface given by,

r = x + y + cos (y − cos x) ˆz (7.1)

representing a zigzag proﬁle having unit amplitudes, and periodicities of 2

. At normal

incidence to the

plane, the polarisation vector of the electric ﬁeld is deﬁned as

ˆx

for TM and p = ˆy for TE polarisations. The normalized surface normal function is

ˆn =

∂

r × ∂

|∂

r × ∂

(7.2)

To induce surface charge density at the interface, the electric ﬁeld will require a

component normal to the surface. The electric vector lying normal to the surface is

then simply,

ˆn

= (ˆn ·p) ˆn (7.3)

The normal electric ﬁeld vector for each polarisation case is shown projected on

to the approximated zigzag surface in ﬁgure 7.2. It is clear that both TM and TE

polarisation vectors result in a surface normal component of electric ﬁeld on the zigzag

surface, and so may induce surface charge.

We can now examine the functional form of the allowed normal electric ﬁeld in the

propagation direction for a SPP. The components,

T E

and

T M

, of the electric ﬁeld

7. Optical Response of Metallic Zigzag Bigratings

T E

T M

0 π 2π

Figure 7.3: The magnitude of the surface normal electric ﬁeld in the x-direction for TM

and TE polarisations.

normal to the surface, lying along the direction of propagation (the

ˆx

direction) and

integrated over y for the two polarisation cases are found to be,

T M

= −

4π sin

(x)

cos (2x) − 3

(7.4)

T E

= −

4π sin (x)

cos (2x) − 3

(7.5)

Both

T E

and

T M

are non-zero, so we may conclude that either polarisation may

induce surface charge and possibly excite SPPs. Plots of

T M

and

T E

in ﬁgure 7.3

show that the electric ﬁeld vector normal to the surface varies spatially twice as fast for

the TM case as for the TE case. This leads to the conclusion that the wavevector of a

TM-coupled SPP is required to be twice that of the TE-coupled case. By expanding

both these expressions as a Fourier sum in

, we may express the

T M

and

T E

as a

sum of plane-waves. It is possible then to ﬁnd which in-plane wavevectors of incident

light are required to match these ﬁeld proﬁles. Doing so yields,

T E

∞

n=1,3,5,...

cos (nx) (7.6)

T M

= a

∞

n=2,4,6,...

sin (nx) (7.7)

7. Optical Response of Metallic Zigzag Bigratings

where

and

are the Fourier series coeﬃcients. For incident light to match

T E

requires a series of only odd-ordered terms, while

T M

requires a series of only even-

ordered terms. Diﬀracted ﬁelds at the surface will contain both odd and even wavevector

components. Equations 7.6 and 7.7 predict that TE polarised light will provide a suitable

electric ﬁeld distribution to enable the excitation of SPPs via only odd-ordered diﬀracted

orders, while TM polarised light will excite SPPs only via even-ordered diﬀracted orders.

This concept will be discussed again with reference to the experimental and modelled

results in sections 7.3.3 and 7.3.4.

We can visualise these coupling conditions in a simplistic diagrammatic way, shown

in in ﬁgure 7.4. Choosing the electric ﬁeld vector to be at a single instance in time, we

consider the eﬀect on the charge carriers under the inﬂuence of this (eﬀectively DC)

ﬁeld. The arrangement of these charges is illustrated in ﬁgure 7.4. These arrangements

lead to an electric potential across the width of the grooves, with zero ﬁeld along the

lines connecting the zigzag apexes (red dots). The component of the respective electric

ﬁelds along the plane of incidence (red arrows) is clearly diﬀerent in both cases, with

the TM ﬁeld distribution varying twice as fast as the TE distribution in the plane of

incidence. Crucially, neither solution for the TE or TM ﬁeld arrangement along the

-axis is zero, meaning that charge density may be induced by either TE and TM

polarisations. Returning to the periodicity of the ﬁelds, it is clear that the TM case

has a wavelength equal to

2, while the TE case has a wavelength equal to

To resonantly drive these ﬁelds with incident light, the impinging ﬁeld must match

the wavevector of these surface plasmon ﬁeld distributions (2k

for the TM case, k

for the TE case). This is achieved via diﬀraction coupling, with ﬁrst order diﬀraction

±k

matching the ﬁelds of the TE case, and the second order diﬀraction,

matching the ﬁeld of the TM case. As such, the polarisation selectivity of coupling

diﬀracted light to the surface plasmons is demonstrated.

7.3.2 Samples

A schematic of the experimental sample is shown in ﬁgure 7.1, comprising of a grating

with a 600

5 nm zigzag pitch (

) to provide diﬀractive coupling to the SPPs. The

periodicity of the non-diﬀracting surface-relief grooves (

) is 150

5 nm. A master of

the zigzag grating was produced in silicon using electron beam lithography, as detailed

in chapter 4. The pattern was exposed in PMMA using a write ﬁeld of 400

m and a

write ﬁeld stitching error on the order of 20 nm. The exposed pattern was developed

and reactive ion etched into a Si master. This produced a 2

5 mm

grating on a silicon

wafer. A scanning electron micrography of one such master is shown in ﬁgure 7.5(a).

Results from two diﬀerent versions of this zigzag grating sample are presented in this

7. Optical Response of Metallic Zigzag Bigratings

−

0 −E

Figure 7.4: Schematic cartoon of light coupling to SPPs on a zigzag grating. The top

cartoons depict a zigzag cavity bounded by two metal zigzag ‘peaks’ (hashed areas)

for a typical zigzag grating as investigated in this chapter. The left images show the

case of TE polarised light, with the TM case on the right. The ﬁeld applied for both

polarisation cases lead to positive (+) and negative (

−

) charge distributions along the

grooves. The component of electric ﬁeld in the

-direction orginating due to these

charge arrangements is shown in red.

7. Optical Response of Metallic Zigzag Bigratings

(a) (b) (c)

Figure 7.5: Scanning electron micrographs of: (a) an example template-strip master in

Si, used for production of zigzag gratings; (b) a polymer replica of a Si master prior to

metallisation, for zigzag gratings embedded in glass; (c) a template stripped sample in

silver.

chapter. One is a zigzag grating produced by the template stripping method detailed

in chapter 4, the SPPs propagating along the air/metal interface. The second sample

is the same zigzag grating embedded in glass, so that the SPPs propagate along the

glass/metal interface. The reason for the two samples is that observation of novel

higher-frequency modes, such as the 3

order diﬀracted SPPs, may be out of the visible

domain when the SPP propagates along the air/metal interface. Using a higher index

bounding material, the phase velocity of the SPPs is reduced and so for the same value

of in-plane momentum, these modes may be brought down in frequency into the visible

domain. The glass/grating sample was produced using a slightly diﬀerent embossing

method, detailed below. In each section, the grating used is clearly stated at the start.

For the glass/grating sample, the structure was duplicated in UV-cured polymer.

Scanning electron microscopy was used to image the surface of this zigzag grating

replica in the polymer (ﬁgure 7.5(b)), prior to metallisation. From this micrograph,

the mark-to-space ratio of the ﬁnal grating structure was determined to be 0.75 with

= 600 ± 5 nm

, and

= 150 ± 5nm

. This pattern was then transferred to an-

other UV cured polymer (Norland Optical Adhesive 73) adhered to a glass substrate

(

632.8 nm

= 1.518

) using an embossing method. Silver was thermally evaporated under

high vacuum (5

−6

mbar) with

50 nm

depositions followed by 10 minutes relaxation

time repeated until an optically thick layer of

200 nm

was recorded on the quartz crystal

thickness monitor. This stepped procedure was used to prevent the temperature rising

on the substrate enough to damage the polymer. The substrate was then adhered to

the back of a prepared glass hemisphere of radius 21.75 mm and refractive index

= 1.518 using index matching ﬂuid, the base of the hemisphere having been modiﬁed

7. Optical Response of Metallic Zigzag Bigratings

in-plane wavevector, k

-1

)

angular frequency, ω (rad s

-1

)

0.5

0.6

0.7

0.8

0.9

1.0

-1

0 0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(a)

in-plane wavevector, k

-1

)

angular frequency, ω (rad s

-1

)

0.0

0.2

0.4

0.6

0.8

1.0

-1

0 0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(b)

Figure 7.6: Experimental data (a) and FEM model prediction (b) of TE polarised

reﬂectivity as a function of in-plane wavevector and angular frequency, mapping SPP

dispersion on a zigzag grating. Blue lines show the positions of diﬀracted light lines

scattered by mk

, where m = ±1, +2, 0

by removing

1 mm

to account for the aﬃxed substrate thickness. This arrangement

allows illumination of the embedded grating structure at the metal/glass interface at all

angles of incidence. The mounting also protects the surface of the silver from sulphur

contamination [129].

7.3.3 Transverse Electric Coupling

The dispersion of the SPPs supported on a zigzag grating is mapped as a function of in

plane wavevector,

, by varying the polar angle

and angular frequency,

, using a

polarised, collimated, monochromatic beam, produced by a white light source together

with a spectrometer. The wavelength range used was 400

< λ

850 nm with the angle

range 7

◦

< θ <

◦

(see chapter 4). When the energy and in-plane momentum of the

light matches that of a SPP mode, the light couples to a SPP and an extremum of the

reﬂectivity is observed. The experimental and theoretical reﬂectivity of incident TE

polarised light at φ = 0

◦

is shown in ﬁgure 7.6, for a zigzag grating in air.

Two dark bands of low reﬂectivity are observed associated with the

±k

light lines

(blue lines). These bands map the position in

ω −k

space of SPPs scattered by grating

vectors equal to

SP P

±k

. As predicted by equation 7.7, this odd-scattered mode

is coupled to by TE polarised light. The dispersion of the mode in the experimental

data (ﬁgure 7.6(a)) and theoretical model obtained using FEM (ﬁgure 7.6(b)) show

excellent agreement. The model parameters used were for a depth of 70 nm and a

100

7. Optical Response of Metallic Zigzag Bigratings

15 20 25 30 35 40

0.4 0.5 0.6 0.7 0.8 0.9 1.0

polar angle, θ (

◦

)

Reﬂectivity

T E

T M

0.8 0.9 1

15 40

Figure 7.7: Specular reﬂectivity of TE (black) and TM (red) polarised light as a function

of polar angle,

, for a wavelength of 632.8 nm,

= 0

◦

. Circles: recorded data with

error bars of 1%, line: ﬁtted FEM model prediction.

mark-to-space ratio between the sub-wavelength grooves of 1, using the silver dielectric

function as reported by Nash & Sambles [

]. The experimental sample is expected

to be shallower than the designed 70 nm from duplication, leading to the observation

that the experimental SPP is not coupled optimally to the incident light, and so the

reﬂectivity band is not as dark as the FEM prediction.

This band of low reﬂectivity is investigated further in ﬁgure 7.7. The reﬂected

intensity of light from a zigzag grating embedded in glass is illuminated with TE

polarised light, at a wavelength of 632.8 nm, and recorded as a function of the polar

angle of incidence,

. At

= 28

◦

, a minimum is observed in the TE reﬂectivity from the

grating close to the ﬁrst-order diﬀraction edge (calculated as

= 18

◦

), indicative of the

excitation of a SPP that has been scattered by one grating vector,

. Incident light

resonantly couples to this diﬀracted SPP and is scattered back in to the specular order

with

phase retardation [

130

]. Out of phase with the specular reﬂection, a dark band is

observed with the missing energy being dissipated as Joule heating. For comparison, the

excitation of an in-plane SPP such as this on a mono-grating or bi-grating would require

101

7. Optical Response of Metallic Zigzag Bigratings

x (nm)

z (nm)

|E| × 10

(V/m)

0 100 200 300 400 500 600

-50 0 50 100 150 200 250

Figure 7.8: The

ﬁeld plot of a SPP excited by TE radiation found in ﬁgure 7.7,

obtained by FEM. The colour scale shows the magnitude of the electric vector,

, for

a cross section of the zigzag grating, in the plane of SPP propagation (

ˆx

). The

arrows show the direction of the electric vector over space. The electric ﬁeld phase

chosen for this travelling wave is arbitrary.

the light to be TM polarised. However, on this zigzag grating for a SPP Bragg scattered

, excitation with TM polarised light is forbidden (see 7.3.1) and no minimum

in the TM data is observed. The data show excellent agreement with the theoretical

reﬂectivity obtained from ﬁnite element method (FEM) modelling, also shown in ﬁgure

7.7, with ﬁtting parameters of

632.8nm

−

5 + 0

, a groove depth of 29.9 nm. The

mark-to-space ratio of 0.75 in the model was determined from the electron micrographs

(ﬁgure 7.5(b)). These ﬁtting parameters conﬁrm that the grating was indeed shallowed

in the fabrication process. Surface roughness and small groove proﬁle changes from the

embossing method are not accounted for in the modelling.

Using the ﬁtted FEM model, the magnitude of the electric ﬁeld in the

plane is

plotted for this SPP mode in ﬁgure 7.8. The electric ﬁeld is evanescently conﬁned to

the surface with the ﬁeld maximum occurring near the top, ﬂat surface of the zigzags

= 29

9 nm and exponentially decaying away into both the air and the metal.

The SPP for this wavelength and angle propagates along the zigzag with a wavevector

given by

SP P

sin θ − k

= 1

, corresponding to a SPP wavelength of

SP P

= 358 nm. The plane of intersection and temporal phase of the wave is arbitrarily

and chosen for the ﬁgure as

= 70 nm and 0

◦

phase, respectively. The propagation of

the SPP along the zigzag, rather than over the grooves, has important consequences for

102

7. Optical Response of Metallic Zigzag Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

−1

0.5

1.5

2.5

3.5

×10

(a)

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

−1

0.5

1.5

2.5

3.5

×10

(b)

Figure 7.9: (a) Experimental data and (b) numerically modelled results of the TE

polarised reﬂectivity as a function of in-plane wavevector and angular frequency, mapping

the SPP dispersion on a zigzag grating in glass. Blue lines show the positions of diﬀracted

light lines scattered by mk

, where m = 1, +2, 0, +3.

the band structure for SPPs on this type of surface, and will be discussed in section 7.4.

It is desirable to fully conﬁrm the coupling conditions outlined in equation 7.7

for other odd-order diﬀracted modes, rather than just the

±k

scattered SPP. By

embedding a zigzag grating in a higher refractive index, such as glass, we are able to

access the +3k

scattered SPP using visible frequency radiation. The experimentally

obtained dispersion of the SPPs on the glass/metal interface is mapped as a function of

in plane wavevector,

, by varying the polar angle

and angular frequency,

, using a

polarised, collimated, monochromatic beam, produced by a white light source together

with a spectrometer, in the wavelength range 400

< λ

850 nm. The reﬂectivity of

incident TE polarised light at φ = 0

◦

is shown in ﬁgure 7.9(a).

With the grating embedded in glass two Bragg-scattered SPP dispersion curves are

observed for this polarisation case. One originates at

−k

and the other at +3

These odd-order diﬀracted SPPs are excited by TE polarised light. There is no evidence

of a SPP dispersion curve from a +2

scattering event as, on a zigzag grating, it

is not excited by TE polarised light as predicted by equation 7.7. Using the ﬁtted

parameters from ﬁgure 7.7, the dispersion plots were reproduced from FEM modelling

and found to be in good agreement and are shown in ﬁgure 7.9(b). The small diﬀerence

between the model asymptotic limits of the SPPs and experiment may be attributed to

103

7. Optical Response of Metallic Zigzag Bigratings

the diﬀerences between the dielectric function of silver used for the model [

] and our

experimental sample.

In summary, the SPPs are shown to be excited on the zigzag grating, with TE

polarised light coupling to the odd-order Bragg scattered SPPs. The dispersion of the

SPPs has been mapped for the two cases of the grating in air and in glass, and agrees

well with a numerical model. No even-order modes are observed when the sample is

illuminated with TE polarised light. There is no excitation of odd-order modes with

TM polarised light.

7.3.4 Transverse Magnetic Coupling

The observation of even-order Bragg scattered SPP modes coupled to by TM radiation

is expected from the allowed charge distributions examined in section 7.3.1. For our

sample, the experimental observation of this mode is diﬃcult, as the zigzag grating is

not optimally coupled to the SPP due to shallowing of the grating during fabrication,

and the even-ordered diﬀraction is also very weak, only mediated by multiple scattering

processes of ±k

To investigate the weak coupling of TM polarised light to the +2

SPP, spectral

plots are analysed for a range of angles for which, theoretically, only the 2

SPP can be

present. A zigzag grating in air was investigated over an angle range of (39

◦

< θ <

◦

to remove the ambiguity of which SPP mode is under observation. Comparison to the

TE reﬂectivity, which should not couple to the SPP, further helps identify the mode.

An example spectrum for both TE and TM polarised illumination at

= 53

◦

is shown

in ﬁgure 7.10. There is a clear broad absorption of the light in the TM case which is

missing from the TE case, reducing the reﬂectivity by approximately 7%.

Both the TE and TM reﬂectivity drop at higher frequencies due to the UV absorption

inherent in the optical response of silver. To remove this high frequency absorption

and better judge the position of the broad reﬂectivity discontinuity associated with

a weakly coupled SPP, the TE and TM reﬂectivity may be normalized with respect

to each other. The TE reﬂectivity spectra, containing no observed SPP modes but

containing the higher frequency absorption of silver, may be used to normalize the TM

data and remove the high frequency silver absorption. This allows a greater accuracy in

the determination of the broad TM coupled SPP mode’s position in frequency, which

is judged by the ﬁtting of a Gaussian curve to the local maximum of the normalized

reﬂectivity in ﬁgure 7.11(a). To further conﬁrm that this reﬂectivity anomaly is due

to the weak excitation of the +2

scattered SPP, the dispersion of this mode in

frequency for the angle range 39

◦

< θ <

◦

is plotted on the modelled dispersion of

the 2

order mode in ﬁgure 7.11(b), obtained from the same FEM model that showed

104

7. Optical Response of Metallic Zigzag Bigratings

angular frequency, ω (rad s

-1

)

reflectivity (a.u.)

2.5 3 3.5 4

× 10

0.8 0.85 0.9 0.95 1

Figure 7.10: An example experimental reﬂectivity plot for TE and TM polarised light

in the visible range at a ﬁxed polar angle of

= 53

◦

. The broad absorption in the TM

case is attributed to the weak excitation of a +2k

scattered SPP.

excellent agreement for the TE case in ﬁgure 7.6. The SPP mode dispersion is found to

be in good agreement with the theoretical prediction for the +2

scattered SPP, and

we conclude that this 2

order mode is excited by TM polarised radiation, and is not

excited by TE polarised light, in agreement with section 7.3.1.

Figure 7.12 shows the dispersion of the SPPs on the zigzag grating embedded in

glass and mapped by using TM polarised light, obtained both experimentally and

using FEM modelling. The only noticeable feature is a band of low reﬂectivity at high

frequency, which does not appear to be associated with any of the in-plane diﬀracted

light lines. This absorption of light is due to the excitation of a SPP that has been

scattered by an out-of-plane grating vector, (

), and is observed as a ﬂat parabolic

curve. The explanation of this SPP band’s shape for out-of-plane scattered SPPs

intersecting the plane of incidence can be found previously in chapter 2. Both the

experimental and theoretical ﬁgures show this band, although the experimental band

lies at a higher frequency compared to the theoretical curve, due to the diﬀerence in the

metal’s dielectric function between that of the experimental sample and the values used

105

7. Optical Response of Metallic Zigzag Bigratings

angular frequency, ω (rad s

-1

)

2.5 3 3.5 4

× 10

0.85 0.9 0.95 1

(a)

in-plane wavevector, k

-1

)

angular frequency, ω (rad s

-1

)

0.75

0.80

0.85

0.90

0.4 0.6 0.8 1 1.2

× 10

3.3 3.4 3.5 3.6

× 10

(b)

Figure 7.11: (a) An example spectral plot and ﬁt at

= 53

◦

for the TM reﬂectivity

normalised to the TE reﬂectivity. (b) comparison between theory (greyscale) and

experiment (black points) for a range of angles (39

◦

< θ < 55

◦

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

−1

0.5

1.5

2.5

3.5

4.5

×10

(a)

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.0

0.2

0.4

0.6

0.8

1.0

−1

0.5

1.5

2.5

3.5

4.5

×10

(b)

Figure 7.12: Experimental data of TM polarised reﬂectivity as a function of in-plane

wavevector and angular frequency, mapping the SPP dispersion on a zigzag grating in

glass. Blue lines show the positions of diﬀracted light lines scattered by

, where

m = 1, +2, 0, +3.

106

7. Optical Response of Metallic Zigzag Bigratings

for FEM. In the experimental case, this frequency is at the limit of the monochromator’s

spectral resolution, and so the signal to noise ratio is lower than for previous plots.

This out-of-plane SPP mode lies signiﬁcantly removed from the diﬀracted light line

for the

±k

light cone, which is not plotted as it occurs at a higher frequency than the

displayed spectral range. Comparing this diﬀerence, ∆

, for in-plane modes observed

under TE illumination, the observation of a

scattered SPP mode with a large enough

∆

to be observed suggests a high level of anisotropic dispersion of the SPP in the

and k

directions. This will be discussed further in section 7.5.

In summary, the observation of weak even-scattered SPPs excited by TM polarised

light has been demonstrated experimentally. To facilitate stronger coupling by even-

order scattered SPPs and TM polarised light, an even

component may to be added

to the zigzag proﬁle, but this would remove the mirror symmetry of the grating and the

polarisation selectivity would be destroyed. This will be discussed further in chapter

8. An out-of-plane SPP associated with

±k

has also been observed, indicating that

the SPP dispersion in the

direction is signiﬁcantly perturbed from the in-plane (

)

dispersion case. This anisotropic propagation of SPPs will be discussed further in section

7.5.

7.4 Band Structure of SPPs on a Zigzag Grating

The formation of SPP band-gaps relies on the ability for a propagating SPP mode to

Bragg scatter and interfere with its counter-propagating self [

]. This interference

forms two possible SPP standing waves on the surface of the grating, which will typically

have two possible ﬁeld solutions, with the ﬁelds of one solution spatially shifted by

with respect to the other. Since only diﬀracted SPPs are observed in the light-cone,

the minimum momentum by which a SPP must scatter to intersect an equivalent

counter-propagating mode, and still be able to couple to light and be observed, is a

total of 2k

On traditional surface-relief gratings, the positions of the standing wave anti-nodes

correspond to induced surface charge on the grating grooves. The two possible standing

waves will cause the surface charge density to sit in diﬀerent arrangements per solution

which will, in general, sit charge in diﬀerent potentials. This means that the energy

of the two standing wave solutions will diﬀer, with no propagating SPPs between the

two energies due to destructive inference. This is the essence of plasmonic band-gap

formation.

On a zigzag grating, the SPPs propagating along the

direction do not run

over surface relief grooves, but follow the zigzag shape along the surface. The two

possible standing waves which form when counter-propagating SPPs interact will diﬀer

107

7. Optical Response of Metallic Zigzag Bigratings

Figure 7.13: An iso-frequency scattergram mapping

-space contours for a 600 nm

zigzag grating for an energy of 2.14 eV (

= 580 nm). The blue circles indicate the

±k

scattered light circles.

in energy by an amount determined by the zigzag structure of the surface and not

the shape or depth of the grooves. The positions of the standing wave anti-nodes

correspond to induced surface charge along the zigzag, and by consideration of these

charge arrangements, the band-gap character of zigzag gratings may be inferred.

7.4.1 Band-Gaps at k

= 0

For a zigzag grating with a periodicity of

= 600 nm in air, the

±k

scattered

SPPs meet at

= 0 within the visible frequency range. An experimentally obtained

iso-frequency contour for these modes is displayed in ﬁgure 7.13. The dark bands of low

reﬂectivity map the SPP mode positions. At

= 0 the +1

and

−

SPP contours

cross, and no signiﬁcant perturbation of these contours is observed. No band-gap is

detected at the mode intersections. The lack of interaction of these modes along

= 0

is consistent with the known scattering properties of a zigzag grating. The grating

proﬁle, being totally described by odd functions, contain no signiﬁcant 2

components.

The result of which is that a SPP must undergo multiple scattering processes of

±nk

where

is odd, in order to diﬀract and intersect other SPP modes. These multiple

scattering processes are very weak, and result in such a small perturbation of the modes

as they intersect in k-space that no band-gap is observed experimentally.

108

7. Optical Response of Metallic Zigzag Bigratings

−

0 +

(a)

−

0 +

(b)

Figure 7.14: Cartoon of the two standing wave solutions for SPPs at

= 0, projected

onto the zigzag surface. Since the anti-nodes of the SPP standing waves lie on the

zigzag midsections in one case (a) and at the apexes in the other case (b), the standing

waves exist in diﬀerent electromagnetic environments and a band-gap may form.

While no band-gap is measured, the existence of a band-gap in this case is not

forbidden. Figure 7.14 shows a diagrammatic representation of the expected locations of

nodes and anti-nodes at

= 0 for the

plane. As previously mentioned, the SPPs on

zigzag gratings run along the zigzag, not over the grooves. For two SPP standing waves

to be energetically dissimilar, the pertinent consideration is the ﬁeld arrangements along

the zigzag structure itself, which is why we consider the

plane. The wavevector of

this standing wave is half that of the Bragg vector by which the SPPs have scattered,

in this case 2

2, leading to a standing wave with the periodicity of

. It is seen

in the ﬁgure that the areas on which the surface charge is induced are at the zigzag

apexes in one case (ﬁgure 7.14(a)), and at their midsection in the other case (7.14(b)).

Using a FEM model, the electric ﬁeld for these two possible standing waves at

= 0

are calculated plotted in ﬁgure 7.15.

The grating is modelled with the following parameters:

= 600 nm

, λ

150 nm

, d

= 40 nm, with a mark-to-space ratio of 1 and a frequency-dependent silver

dielectric function [

]. We may identify if these standing waves diﬀer in energy, and

which of these two is higher in energy relative to the other, by the ﬁeld decay length into

the two media. The higher-energy solution, having been shifted higher in frequency and

therefore towards the diﬀracted light line, will be more photon-like than the lower energy

solution, its ﬁelds extending further into the dielectric than the low-energy solution. The

lower-energy solution will have been shifted lower in frequency, away from the light line

109

7. Optical Response of Metallic Zigzag Bigratings

x (nm)

y (nm)

0.1

0.2

0.3

0.4

0.5

|E| (V/m)

0 300 600

75 150

(a)

x (nm)

z (nm)

0.05

0.10

0.15

0.20

0.25

0.30

|E| (V/m)

0 100 300 500

0 100 200 300 400

(b)

x (nm)

y (nm)

0.1

0.2

0.3

0.4

0.5

|E| (V/m)

0 300 600

0 75 150

(c)

x (nm)

z (nm)

0.1

0.2

0.3

0.4

|E| (V/m)

0 100 300 500

0 100 200 300 400

(d)

Figure 7.15: The magnitude of electric ﬁeld for the SPP standing waves at

= 0. (a-b)

The high-energy solution for (a) the

plane at

= 41 nm and (b) the

plane, for

= 75 nm. (c-d) The low-energy solution for (c) the

plane at

= 41 nm and (d) the

xz plane, for y = 75 nm.

and so will be more plasmon-like, constrained closer to the surface. These deductions

can be applied in the comparison between the ﬁelds of the high energy standing wave in

ﬁgure 7.15(b) and the lower energy standing wave in ﬁgure 7.15(d). Subﬁgures 7.15(a)

and 7.15(c) show the magnitude of electric ﬁeld for an arbitrary phase in a plane placed

1 nm above the zigzag surface at

= 41 nm. The two ﬁeld distributions have the

expected period of

. The higher-energy standing wave (ﬁgure 7.15(a)) has the ﬁeld

maxima along the edges of the zigzag grooves, while the lower energy standing wave

has the ﬁeld hotspots on the zigzag apexes. It is this energy diﬀerence between the two

possible standing waves which leads to the plasmonic band-gap.

Analogous to the case of surface relief gratings, whereby the charge sits at the

110

7. Optical Response of Metallic Zigzag Bigratings

x (nm)

y (nm)

0 300 600

0 75 150

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

-1.5 -1 -0.5 0 0.5 1 1.5

× 10

1.95 1.96 1.97 1.98 1.99 2

× 10

(a)

x (nm)

y (nm)

0 300 600

0 75 150

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

-1.5 -1 -0.5 0 0.5 1 1.5

× 10

1.92 1.93 1.94 1.95 1.96 1.97

× 10

(b)

Figure 7.16: Manipulation of the band-gap observed at

= 0 through the increasing

of zigzag amplitude. The unit cell and corresponding SPP dispersion around the

intersection of the +1

and

−

scattered SPPs are shown for (a) The zigzag

grating presented in this chapter and (b) a theoretical zigzag with a larger zigzag

amplitude.

maxima or minima of the grooves (outlined in the background theory chapter 2), we

expect the dependence of this band-gap size to vary as some function of zigzag (in the

surface plane) amplitude.

An eigenmode solution is found for two possible unit-cells using FEM modelling,

the results of which are shown in ﬁgure 7.16. The parameters for this model used the

unit cells shown, with depths of 40 nm, silver parameters from literature [

] and with

the global environment refractive index

= 1

518. The large-amplitude zig-zag was

produced by shifting the central apex in the

direction by 150 nm. This causes multiple

zigzags to overlap in a single unit cell, in order to maintain the sub-wavelength period of

= 150 nm. It is clearly shown that with an increase of zigzag amplitude, a band-gap

is opened at

= 0. This gap is small, as the SPPs must still rely on multiple scattering

processes to diﬀract and interact. This is the ﬁrst example in this thesis of using the

zigzag structure and symmetry to manipulate the band structure of the SPPs, a topic

111

7. Optical Response of Metallic Zigzag Bigratings

in-plane wavevector, k

−1

)

angular frequency ω ( rad s

−1

)

-1

0.51 0.52 0.53

× 10

2.8 2.82 2.84

× 10

Figure 7.17: Modelled SPP dispersion around the intersection point of the

−

and

+2k

scattered SPPs meeting at the ﬁrst BZ boundary (red dashed line).

which will be examined further in chapter 8.

7.4.2 Band-Gaps at the First Brillouin Zone Boundary

Normally at Brillouin zone (BZ) boundaries, diﬀractive coupling results in two counter

propagating SPPs, which establish a standing wave [

]. The two standing wave solutions

correspond to diﬀerent ﬁeld distributions with respect to the grating proﬁle, and between

these two energy solutions no surface modes propagate: a SPP band-gap forms. However,

the experimentally mapped dispersion shown in ﬁgures 7.6 and 7.9(a), and also the

predictions from FEM modelling (ﬁgures 7.6 and 7.9(b)) shows no measurable SPP

band-gaps at the ﬁrst BZ boundaries for this zigzag grating.

There are two considerations to be made here to determine the possibility of a band-

gap between the

−k

and +2

scattered SPPs: (1) is there a suﬃcient scattering

amplitude present for these modes to Bragg scatter and interact with one another?, and

(2) how do the possible standing wave states diﬀer in energy to produce a frequency

band of disallowed SPP propagation, if at all?

Firstly, for the

−k

and +2

SPPs modes to scatter and interact strongly, they

require a total momentum of 3

. If the grating were to contain a 3

component in

it’s surface proﬁle, this would be a simple direct-scattering event and a strong interaction

can occur. Since we observe the 3

scattered SPP in the reﬂectivity mapped in ﬁgure

7.9(a) (interacting with zero-order light), we can safely assume that the grating provides

112

7. Optical Response of Metallic Zigzag Bigratings

−

0 +

(a)

−

0 +

(b)

Figure 7.18: Cartoon of the two standing wave solutions for SPPs at the 1

BZ. Since

the peaks and troughs of the zigzag exist in equivalent electromagnetic environments,

the solutions are degenerate in energy, and no band-gap forms.

a strong 3

scattering mechanism. However, despite the ability for the grating to

scatter SPP modes with a reasonable 3

component, there is still no observable

band-gap between the

−k

and +2

at the ﬁrst BZ boundary. A numerical model

of these modes crossing at the BZ is shown in ﬁgure 7.17. The numerical model is

used to extract the possible eigenmodes of the SPPs at the 1st BZ boundary, with

modelling parameters of

= 600 nm

, λ

= 150 nm

, d

= 40 nm

a mark-to-space ratio

of 1 and

(

) from literature [

]. No band-gap is observed, to within the accuracy of

the numerical model.

The lack of band-gap at the ﬁrst BZ boundary is explained considering the allowed

standing-wave symmetries on a zigzag grating. Generally, the in-plane wavevector of

a standing SPP wave is half that of the total Bragg vector by which the two counter

propagating SPPs have been scattered. For the

−k

and +2

scattered SPPs,

crossing at the ﬁrst BZ boundary, the SPP wavevector is 3

2. Through symmetry,

there are two possible solutions for this standing wave on a zigzag grating, with the

areas of high ﬁeld of one solution shifted spatially by

4 with respect to the other

solution. These two solutions are drawn in ﬁgure 7.18, illustrating the node and anti-node

positions of a standing wave along the zigzag. Since SPPs run along the zigzag pattern,

(rather than over the grooves), the only possible manner by which to organise charge

in dissimilar energetic arrangements is to consider how the charge may accumulate

on the zigzag groove sides. Comparing this simple cartoon with the magnitude of

electric ﬁeld in the

plane, 1 nm above the grating surface, shown in ﬁgure 7.19, we

113

7. Optical Response of Metallic Zigzag Bigratings

x (nm)

y (nm)

0.1

0.2

0.3

0.4

|E| (V/m)

0 300 600

0 75 150

(a)

x (nm)

y (nm)

0.1

0.2

0.3

0.4

|E| (V/m)

0 300 600

0 75 150

(b)

Figure 7.19: The magnitude of electric ﬁeld,

|E|

for the degenerate SPP standing waves

at the ﬁrst BZ. (a) One solution for the

plane at

= 41 nm and (b) The second

solution for the xy plane at z = 41 nm.

see that the unit cell contains three hotspots, corresponding to the expected period

SP P

= 2

π/|k

SP P

= 2

3, with the ﬁeld arrangements sitting in the locations

predicted by the cartoon. These ﬁeld plots are extracted from the same numerical model

that mapped the dispersion in ﬁgure 7.17. In both cases, one hotspot sits on a zigzag

apex, and the other two sit symmetrically either side on the zigzag edges.

The electric ﬁeld arrangements for the two possible standing wave modes at the BZ

boundary in the grooves (in the

plane with

= 20 nm) and in the

plane (with

= 75 nm), are extracted from the numerical model and plotted in ﬁgure 7.20. These

standing waves have three ‘hotspots’ per unit cell, corresponding to a standing wave

with the expected period of

SP P

= 2

3. Comparing the magnitude of electric ﬁeld

in the grooves for the

plane in ﬁgures 7.20(b) and 7.20(d), it is clear that the ﬁeld

arrangements are shifted spatially with respect to each other by

4, as one would

expect for two standing wave solutions which in general will lie

π/

2 degrees out-of-phase

with each other. Figures 7.20(a) and 7.20(c) show the calculated ﬁeld arrangements as

ﬁgure 7.18 illustrated. Lying at the same potential, these ﬁeld arrangements are clearly

equivalent, shifted spatially by

with respect to each other, yet both occupying a

degenerate electromagnetic environment. The magnitude of electric ﬁeld in the

plane

presented in ﬁgures 7.20(b) and 7.20(d) show that the decay of the electric ﬁelds into

both bounding media are the same for each case, and that neither solution can be

labelled more ‘photon-like’ nor ‘plasmon-like’ than the other. This further demonstrates

the equivalence in energy of these two standing waves.

The equivalence in energy of these two standing wave solutions can be broken by

114

7. Optical Response of Metallic Zigzag Bigratings

x (nm)

y (nm)

0.02

0.04

0.06

0.08

0.10

|E| (V/m)

0 300 600

0 75 150

(a)

x (nm)

z (nm)

0.02

0.04

0.06

0.08

0.10

0.12

0.14

|E| (V/m)

0 100 300 500

0 100 200 300 400

(b)

x (nm)

y (nm)

0.02

0.04

0.06

0.08

0.10

0.12

|E| (V/m)

0 300 600

0 75 150

(c)

x (nm)

z (nm)

0.05

0.10

0.15

|E| (V/m)

0 100 300 500

0 100 200 300 400

(d)

Figure 7.20: The magnitude of electric ﬁeld for the degenerate SPP standing waves at

the ﬁrst BZ. (a-b) One solution for the (a)

plane at

= 20 nm and (b) the

plane,

for

= 75 nm. (c-d) The second solution for the (c)

plane at

= 20 nm and (d) the

xz plane, for y = 75 nm.

the removal of the zigzag grating’s mirror plane. By making the zigzag asymmetric,

the electric ﬁeld arrangements for both possible standing waves diﬀer in energy and a

band-gap may form. This is demonstrated in chapter 8

7.5 Anisotropic Propagation of SPP Modes for Self Colli-

mation

Plasmonic circuits couple and direct SPPs along designed structures to provide a transfer

of power and information [

131

]. These optoelectronic devices require the development

of SPP optics which behave like their classical optics counterparts.

115

7. Optical Response of Metallic Zigzag Bigratings

One of the most crucial roles of these surface-optics is the ability to generate

a collimated SPP beam, so that the SPPs may be directed eﬃciently without any

detrimental loss across the device due to divergence. The ideal device should receive

incident SPPs over a wide range of angles and the SPPs should then propagate through

the component with a narrow angular divergence. This eﬀect has been observed for

light in photonic crystals, and was named self-collimation. [132, 133]

Recently, such components have been manufactured that use square bi-gratings to

manipulate the local curvature of the SPP iso-frequency contour. These gratings have

demonstrated self-collimation in the microwave regime [

134

] as well as in the visible

[

]. They have also been designed to achieve all-angle negative refraction of surface

waves [

135

136

]. The work by Stein et al. [

] showed that the self-collimation eﬀect

could be achieved around the M symmetry point in

-space due to the mini-gaps formed

by band-gapping SPPs over a momentum range of 1

−1

. It is desirable that this

momentum range is large, so as to maximise the angular divergence over which the

device could collimate SPPs.

We have found that zigzag gratings possess a mechanism for the collimation of SPPs

at a single wavelength. This mechanism is similar to previous work, as it relies on

plasmonic band-gaps to cause anisotropic propagation of SPPs along the surface, which

is observed as a deformation of the SPP iso-frequency contours to approximately ﬂat

bands with no curvature. However, the azimuthal angle range over which SPPs possess

a single direction is far larger than previously reported results, extending over the entire

observable region of k-space.

Up until now, we have only considered the eﬀects of the

periodicity of the zigzag

grating with respect to the SPP propagation. With a designed orthogonal pitch of

= 150 nm, the diﬀracted SPPs from a

scattering mechanism have such a large

momentum that for the plane of incidence

= 0

◦

, the SPP features will lie well above

the visible frequency range.

However, an important point to note about SPPs travelling along the

direction

is that they will be travelling over surface relief grooves and not, like the

±k

surface

plasmons, along the zigzag contours. This means that the charge arrangements for

standing-waves at the BZ boundaries will diﬀer in energy considerably, as for traditional

surface-relief gratings. Plasmonic band-gaps in the

direction are expected, and

will be similar to those on typical bigratings. Evidence for these band-gaps are seen in

ﬁgure 7.21. The experimental results show the dispersion mapped using the reﬂectivity

of the zigzag grating to indicate the SPP mode positions for a wavelength range of

450

< λ

800 nm and an angular range 7

◦

< θ <

◦

, for two planes of incidence. One

plane of incidence includes the

grating vector (

= 0

◦

) and the orthogonal plane of

incidence (

= 90

◦

) contains the

vector. In both cases the

scattered SPP is

116

7. Optical Response of Metallic Zigzag Bigratings

in-plane wavevector, k

-1

)

angular frequency, ω (rad s

-1

)

0.0

0.2

0.4

0.6

0.8

1.0

-1

0 0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(a) R

at φ = 0

◦

in-plane wavevector, k

-1

)

angular frequency, ω (rad s

-1

)

0.0

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4

× 10

(b) R

at φ = 90

◦

Figure 7.21: Experimentally obtained dispersion plots for a zigzag grating at

= 0

◦

and φ = 90

◦

observed. For

= 0

◦

the SPP dispersion is an asymptotic curve, while for

= 90

◦

is seen as a hyperbolic conic intersection, but both map the same SPP in momentum

space. For the coupling of light to the SPP in ﬁgure 7.21, the polarisation of the incident

light has been rotated 90

◦

, and so is TM polarised.

It is found that in the

= 90

◦

orientation, the SPP dispersion curve asymptotes far

faster than the

= 0

◦

case. This lower asymptote is evidence of a large band-gap at

the ﬁrst BZ in the

direction, with the mode meeting the BZ boundary with zero

group velocity at 2

−1

. This band-gap is the result of the interaction between

the

±k

and

±k

scattered SPPs which are separated by a single scattering

vector that is also a harmonic of the grating surface proﬁle,

. Strong interaction is

then expected, and the freedom of the charge to organise into energetically dissimilar

arrangement in the grooves leads to a large plasmonic band-gap in this direction.

The relevance of this large band-gap to the collimation of SPPs becomes apparent

in the evolution in wavelength of the SPP iso-frequency contours mapped using imaging

spectrometry, shown in ﬁgure 7.22.

The scattergrams map the iso-frequency contours of the

scattered SPPs

-space, with the minimum of reﬂection indicating the SPP mode position as the

reﬂection is suppressed by SPP excitation. The polarisation is set so that the electric

ﬁeld lies orthogonal to the

vector at

= 0, satisfying the TE excitation condition

for these SPPs in this plane. As detailed in section 7.3.3, the

scattered SPPs are

not excited with this polarisation, and are not observed. The grating used for these

scattergrams is a silver zigzag grating in air as detailed in section 7.3.3, with a depth of

d ≈ 40 nm.

117

7. Optical Response of Metallic Zigzag Bigratings

(a) 650 nm (b) 600 nm (c) 580 nm

(d) 550 nm (e) 500 nm (f) 450 nm

Figure 7.22: Measured iso-frequency contours of a zigzag grating for a range of wavelengths. The blue circles indicate calculated

diﬀraction edges.

118

7. Optical Response of Metallic Zigzag Bigratings

Figure 7.23: The SPP iso-frequency contours mapped using imaging scatterometery at

= 450 nm. The green arrows show the direction of the SPP group velocity along

the contour, indicating that for this wavelength the SPP waves only travel in the

±k

direction. The red dotted lines show the position of the BZ boundary.

As the frequency increases (the wavelength shifting from red to blue), the

scattered SPP contours grow increasingly ﬂat-banded. This is due to the increasing

overlap and interaction with the

scattered SPPs and the formation of

a large band-gap well outside the available momentum space to the free-space light.

= 450 nm, the SPP bands become ﬂat, with essentially no curvature. This

scattergram is repeated in ﬁgure 7.23, with some additional important annotations.

Since the group velocity of the SPP wave is determined from

∇ω

(

), the

direction of the SPP energy ﬂow is in the direction orthogonal to the SPP contour (see

chapter 6 for more details on this inference). Six example group velocity directions are

shown as green arrows in ﬁgure 7.23, showing that, for a given scattered SPP, the group

velocity is only in the

direction. This corresponds to a self-collimated SPP wave

across a momentum range of at least 2

−1

, an order of magnitude higher than

previous results [

], and successfully collimates SPPs for an incident azimuthal range

of −63.4

◦

< φ < 63.4

◦

at λ

= 450 nm.

An Important observation is to be made of ﬁgure 7.23; that the

scattered

SPP contours lie outside of the BZ boundary (red dotted lines). Two comments must be

made about this: the ﬁrst is that the BZ represents the smallest unit cell which, when

repeated by translational symmetry operations, reproduces fully the band structure of

the system. Clearly, the BZ lying between the two red lines does not replicate the band

structure of the grating once translated. This is because the

scattered SPPs

119

7. Optical Response of Metallic Zigzag Bigratings

are not coupled strongly to light in our experiment. The eignestates for the SPPs are

still present in the BZ, just not excited by the incident radiation, and so the validity of

the BZ holds. The second point is that the

SPPs have passed through the BZ

boundary unperturbed. This is important, as if the contours were perturbed at this

boundary, the eﬃciency of the collimation of SPP waves would be impacted. Because

band-gaps at the ﬁrst BZ are forbidden (not just weak) in the

direction, unwanted

perturbation of the SPP contour at this boundary will not occur. This is a condition

due to the mirror symmetry of the zigzag grating surface, and an analogy cannot be

found in traditional gratings.

7.6 Conclusions

This chapter has introduced a new type of diﬀraction grating, a zigzag grating. This

grating uses sub-wavelength surface structure to provide a diﬀractive periodicity to

wavelength-scale light. SPPs may be diﬀractively coupled to using a metallic zigzag

grating, and their excitation and band structure are found to depend on the symmetry

of the zigzag pattern.

The polarisation of light coupling to SPPs on such a grating is found to be dependent

on the diﬀracted order used. For odd-ordered diﬀraction, TE polarised light couples to

the SPPs, while for even-ordered diﬀraction, TM polarised light is required. This has

been explained using a simple analytical formula which considers the available normal

component of electric ﬁeld to the surface of the grating, and the polarisation selectivity

has been demonstrated experimentally on fabricated silver zigzag gratings.

This polarisation selectivity may be found to be of use for plasmonic devices in

which polarisation separation is desirable. As an example, the incorporation of zigzag

gratings in metal-insulator-metal structures designed for the generation of light could

yield a polarisation dispersing light source. These SPP mediated light sources, when

incorporated with a zigzag grating, would emit diﬀerent polarisations of light in to

diﬀerent diﬀracted orders. Breaking the mirror symmetry of the zigzag pattern provides

a route to polarisation-independent absorption of light into SPPs, a topic which is

explored in greater detail in chapter 8.

The band structure of SPPs on zigzag gratings is found to be highly dependent on

the surface symmetry. Most strikingly, the formation of a band-gap at the ﬁrst BZ

boundary is forbidden by the degeneracy of the allowed standing wave states.

Using scatterometry, the propagation of SPPs on such surface is shown to be highly

anisotropic, due to the large band-gaps which occur orthogonal to the diﬀraction plane.

These highly perturbed SPP contours, combined with the forbidden band-gaps at

the ﬁrst BZ boundary in the

-direction, lead to wide-angle surface wave collimation.

120

7. Optical Response of Metallic Zigzag Bigratings

The use of zigzag gratings to generate surface-waves with highly directional planar

wave-fronts in plasmonic circuits, over a wide range of azimuthal angles, could be of

great interest to optoelectronic engineers. Further investigation of this phenomenon is

left for future work, with the recommendation of using near-ﬁeld imaging of SPPs to

characterise the surface waves and exploring also the collimating eﬀect of the grating.

121

Chapter 8

Optical Response of Asymmetric

Zigzag Bigratings

8.1 Introduction

This chapter demonstrates how the surface symmetry of a zigzag grating may be used

to manipulate both the coupling of light to SPPs, and the band structure of SPPs on

the zigzag surface. Much of the discussion in this chapter draws parallels with the

zigzag grating explored in chapter 7. The zigzag grating explored in chapter 7 will be

referred to as a ‘symmetric’ zigzag grating, due to the presence of mirror symmetry in

the

plane. The zigzag grating which shall be the topic of this chapter has no such

symmetry, and so will be referred to as an ‘asymmetric’ zigzag grating. The asymmetric

zigzag grating is formed from a set of sub-wavelength (non-diﬀracting) grooves that are

zigzagged along their length such that the zigzag apexes do not lie at high symmetry

points within the rectangular unit cell. The length scale of the zigzag pitch is on the

order of the wavelength of impinging light, so that the grating may diﬀractively couple

to SPPs.

The resulting structure is 2D chiral, much like ‘ﬁsh-scale’ nanowires reported previ-

ously, [

137

–

141

] but we restrict our investigation to the excitation and band structure

of SPPs along such a surface, rather than any optical chirality exhibited by the grating.

In this chapter it is demonstrated that any polarisation of light may couple to the

SPP modes on a zigzag grating which possesses no mirror symmetry. Furthermore, it

is shown that light of diﬀerent polarisation will couple to the exact same SPP modes,

and these SPP modes propagate in the same direction, regardless of polarisation. This

latter point is subtle, but important, as previous work has shown that the excitation

of SPPs on crossed bigratings may also couple arbitrary polarised light into SPPs on

the surface [

142

], but in this case diﬀerent incident polarisations states excite

122

8. Optical Response of Asymmetric Zigzag Bigratings

diﬀerent, or multiple, SPPs. The coupling of arbitrarily polarised light to SPPs on deep

lamella gratings has also been shown [

143

], where the SPP modes evolve to become

similar in character to localised cavity modes. In the zigzag case presented here, light

of any arbitrary polarisation is coupled to the same SPP modes, and the energy ﬂow

across the surface is mediated by these SPPs travelling in a single direction only. This

makes these asymmetric gratings an excellent possible component for eﬃcient plasmonic

circuitry [131].

The manipulation of available Fourier components by altering the geometry of the

asymmetric zigzag surface also allows the design of the SPP band structure on such

gratings. We ﬁnd in section 8.4, theoretically and experimentally, how the asymmetric

zigzag pattern can cause band gaps to form between the +

and

−k

scattered SPPs,

by providing a direct scattering mechanism by which they may interact. We also show

in section 8.4 how the degeneracy of standing wave solutions at the ﬁrst BZ boundary,

described in chapter 7 for the symmetric case, is destroyed in the asymmetric zigzag

case. The coupling of light to the two energetically dissimilar standing wave states is

also investigated, as the high energy solution is found to be poorly coupled to light

compared to the lower energy mode.

Finally, highly anisotropic propagation of SPPs along zigzag gratings, when combined

with the possible large band gap at the ﬁrst BZ boundary, causes the formation of a

full-plasmonic band gap, for which propagation of SPP for a given frequency range is

forbidden in all directions. Full plasmonic band-gaps have shown potential in surface

plasmon based lasers [

–

144

], and are usually only found in systems with hexagonal

[114], not rectangular, symmetry.

8.2 The Asymmetric Zigzag Grating

The coordinate system used for the asymmetric zigzag grating is shown in ﬁgure 8.1.

Light impinges on the surface at a polar angle

in the plane of incidence that lies

at an angle

, with

= 0

◦

containing the primary grating vector,

= 2

x/λ

The polarisation of the light is deﬁned as TM polarised light when the electric vector

is in the plane of incidence, and TE polarised light when the electric vector lies

perpendicular to the plane of incidence. The oﬀset of the zigzag apex,

is the distance

between the centre of the unit cell and the closest apex, as shown. It is this oﬀset

that removes the mirror symmetry of the zigzag grating. The grating grooves are of

a depth

. The target parameters (for fabrication) used in this chapter are as follows:

= 600 nm, λ

= 150 nm, d ≈ 40 nm

and

δ = 150 nm

with the grating made of silver.

The zigzag amplitude is deﬁned as the distance between a low and high apex of the

zigzag divided by two.

123

8. Optical Response of Asymmetric Zigzag Bigratings

T M

T E

plane of incidence

Figure 8.1: Coordinate system for the asymmetric zigzag grating. The experimental

parameters were λ

= 600 nm, λ

= 150 nm, d ≈ 40 nm, and δ = 150 nm .

The zigzag grating presented in this chapter has a reduced symmetry from the zigzag

grating presented in chapter 7. This is achieved by oﬀsetting one of the zigzag apexes

along

with respect to the centre of the unit cell by a distance

= 150 nm, as shown in

ﬁgure 8.2. This produces a grating structure with no mirror or rotational symmetry in

real space, but still having a rectangular unit cell and a rectangular lattice in reciprocal

space. For ease of description, we refer to the ‘left hand side’ of the grating, the region

0 nm < x < 450 nm

(to the left of the apex), as ‘region 1’ and the right hand side of

between 450 nm < x < 600 nm as ‘region 2’.

The samples are produced using electron beam lithography as detailed in chapter 4.

Scanning electron micrographs of the silicon master and the template stripped sample

in silver are shown in ﬁgure 8.3, with the following parameters:

= 597

5 nm

, λ

156 ± 9 nm, d ≈ 40 nm, δ = 126 ± 5 nm with a zigzag amplitude of 121 ± 3 nm.

Due to the template stripping method of fabrication, the produced silver grating

is an inverse duplicate of the grating master. The master SEM 8.3 also shows a clear

stitching error on the right of the SEM, where the zigzag continuity has been broken.

These stitching errors are infrequent and, over the area of the grating, do not eﬀect the

optical response greatly.

Notice that the zigzag amplitude of the sample is greater than that of the unit

cell in ﬁgure 8.2. The experimental and modelled results will show that since an

124

8. Optical Response of Asymmetric Zigzag Bigratings

x (nm)

y (nm)

0 300 600

0 75 150

(a) Symmetric

x (nm)

y (nm)

0 300 600

0 75 150

0 150

region 1 region 2

300 + δ

(b) Asymmetric

Figure 8.2: Two possible unit cells for a zigzag grating. (a) The grating unit cell

explored previously in chapter 7. (b) An asymmetrical zigzag grating, with the central

apex of the zigzag shifted by 150 nm. This shift removes the mirror-symmetry of the

zigzag.

(a) Master (b) Metal grating replica

Figure 8.3: Scanning electron micrographs of (a) The asymmetric zigzag silicon master

and (b) the template stripped sample in silver.

125

8. Optical Response of Asymmetric Zigzag Bigratings

increased zigzag amplitude alone does not alter the symmetry of the zigzag surface, the

fundamental optical response of the grating is unaltered. For simplicity in understanding

only consideration of the unit cell in ﬁgure 8.2 is necessary to gain an insight into the

optical response of our sample.

8.3 The Coupling of Polarised Light to SPPs on an Asym-

metric Zigzag Grating

8.3.1 Theory

In chapter 7, it was shown that a zigzag grating possesses a surface proﬁle that allows

surface charge density oscillations to be induced by either TE or TM polarised light. This

is because both TE and TM light provide a component of the incident electric ﬁeld that

lies normal to some part of the metal surface. For the case of the grating examined in

chapter 7, the ﬁeld proﬁles of these induced charge density oscillations were constrained

by the symmetry of the surface so that the wavevector of SPPs coupling to TE polarised

light are only the odd-orders of diﬀraction, while the TM polarised light could only

match the wavevector of even-order diﬀracted SPPs. These coupling conditions can

be deduced from the plane wave expansion equations 7.4 and 7.5. Since the functions

T E

and

T M

will possess the same mirror symmetry as the surface function, so the

Fourier expansions of

T E

and

T M

only require every-other Fourier coeﬃcient to fully

describe the functions. This is what leads to the polarisation selectivity of this type of

grating.

However, by removing the mirror symmetry of the zigzag surface the equivalent

expressions for

T E

and

T M

in the asymmetric zigzag case will themselves not be

mirror symmetric. We can demonstrate this simply by considering the E ﬁeld for normal

incident light, as shown diagrammatically in ﬁgure 8.4.

In this ﬁgure, the possible electric ﬁelds across the zigzag grooves are shown for

two electric vectors, one in the

direction (left), and one in the

direction (right).

The electric vectors pointing along a single axes like this represents two orthogonal

polarisations at a single instant in time when the light in incident normal to the surface.

The plane of incidence at

= 0

◦

is not deﬁned, as the wavevector possesses no component

in the surface plane

, but for

= 0

◦

and for a very small polar angle

, the polarisation

is deﬁned as TE for the vector lying along the

axis (right) and TM for the vector lying

along the

axis (left), so we shall refer to the left-hand diagram as the TE polarised

case, and the right-hand side as the TM polarised case.

The ﬁgure shows that for the electric vector oriented along the

or the

axis, the

incident ﬁeld may provide a normal component to the grating surface and so induce

126

8. Optical Response of Asymmetric Zigzag Bigratings

T M

T E

0 0

Figure 8.4: (above) A schematic of the electric ﬁeld lines in the grooves (black arrows) of

an asymmetric zigzag grating for two polarisation cases, and the resulting

component

(red arrows). The points connecting the apexes lead to ﬁxed points of zero

, leading

to an asymmetric ﬁeld distribution along the plane of propagation (bottom).

surface charge. Because of the surface geometry, charge density oscillations may be

induced on the grating surface and so there exists a possibility of exciting SPPs. Since

in our experiments

is sub-wavelength, we need only consider the electric ﬁeld proﬁle

along the

axis, as this is the axis along which momentum conservation requires

diﬀracted SPPs to propagate. The

-components of the electric ﬁelds in the grooves

are shown as red arrows in the ﬁgure. Possible example functions for each polarisation

which fulﬁl the asymmetry criteria of the grating are shown in the lower half of ﬁgure

8.4, which have been deduced from the diagrams above. These functions are neither odd

nor even, both possessing only the symmetry of the grating (a rotation and translation

operation or ‘glide operation’), and will in general vary in amplitude in the two regions.

The key point illustrated in this ﬁgure is that the normal component of electric ﬁeld

varies in the

-direction asymmetrically with respect to the unit cell. This is because the

-component of the electric ﬁeld originating from charge carriers must be pinned to zero

at the zigzag apexes by the geometry of the grating, and the position of these apexes

do not lie at points of any high-symmetry. The plane wave expansion description of

such general diﬀracted SPPs ﬁelds on such a surface will necessarily contain all Fourier

127

8. Optical Response of Asymmetric Zigzag Bigratings

components including both odd and even harmonics, for both polarisation cases.

In chapter 7, we calculated the Fourier series of

T E

(

≡ E

T E

) and

T M

(

≡ E

T M

)

by ﬁrst deriving the electric ﬁeld from the expected induced surface charge along

the functions

T E

and

T M

. Examination of the plane wave expansion of these ﬁelds

determines the coupling of light to diﬀerent orders of diﬀracted SPP. In the asymmetric

case, the strict derivation of

T E

and

T M

is diﬃcult, due to the lack of symmetry

in the grating’s geometry. We can instead construct an example function from careful

consideration of the conditions an

ﬁeld must possess along such a grating. We place

ﬁve constraints on any function we choose: (1) That the function must be continuous

along

, with no sharp discontinuities or undeﬁned points; (2) that the function is

periodic, with a period equal to that of a single grating period,

; (3) that the function

must equal

= 0 at the

coordinates which correspond to a zigzag apex; (4) that

between these zeros, the magnitude of

will reach a maximum value determined by

the polarisation and local geometry and; (5) the resulting function must possess the

same symmetry as the grating (a single C

rotation operation).

Such a function must be deﬁned piecewise, with the same symmetry as the intuitive

example functions drawn in ﬁgure 8.5. Choosing a simple sine wave for each region, the

function we shall use is deﬁned as,

T M

(x) =







T M∗

(1 + sin (−

2πx

(L+δ)

)) 0 < x < L + δ

T M∗

(−1 + sin (

2π(x−L−δ)

(L−δ)

)) L + δ < x < 2L

(8.1)

T E

(x) =







T E∗

(1 + sin (−

2πx

(L+δ)

)) 0 < x < L + δ

−E

T E∗

(−1 + sin (

2π(x−L−δ)

(L−δ)

)) L + δ < x < 2L

(8.2)

Where 2

. The variable

represents a positive oﬀset of the central zero point

from

and the function is deﬁned as periodic with a period 2

. These functions

are plotted in ﬁgure 8.5

The magnitude of

in the two regions (

T E∗

, E

T E∗

, E

T M∗

, E

T M∗

) will not in

general be equal and will vary as a function of the oﬀset

. This magnitude will be a

result of the dot product between the impinging ﬁeld vector and the local surface normal,

which will be diﬀerent for TE and TM polarised light. To a good approximation, the

amount of induced surface charge density will be proportional to this dot product, and

the resulting SPP electric ﬁeld may be decomposed along the

axis to see which plane

wave components will couple strongly to it. Using these considerations the expressions

128

8. Optical Response of Asymmetric Zigzag Bigratings

-1.0 -0.5 0.0 0.5 1.0

0 L 2L

L L + δ

T E

T M

Figure 8.5: (a) The piecewise function representing

for TE (

T E

) polarisation (black

line) and TM (

T M

) polarisation (red dotted line). Blue background shows the zigzag

grating in the

plane, with the

-coordinates aligned with the plot.

= 1 in this

diagram.

for maximum |E

| will be given by,

T E∗

= E

cos



tan

−1



L − δ



sin



tan

−1



L − δ



(8.3)

T M∗

= E

sin



tan

−1



L − δ



(8.4)

T E∗

= E

cos



tan

−1



L + δ



sin



tan

−1



L + δ



(8.5)

T M∗

= E

sin



tan

−1



L + δ



(8.6)

where

is half the zigzag amplitude and

is the initial ﬁeld magnitude. These

functions are plotted in ﬁgure 8.6 as a function of

, and show how the

component

varies rapidly for region 2, as the increasing oﬀset forces the normal component of the

surface increasingly in the

direction. In the extreme case of

, TM polarised

light will be perpendicular to the region 2 surface normal, and TE will be parallel to it,

leading to E

T M∗

= E

and E

T E∗

= 0.

We may now expand these two functions as a Fourier sum to determine which

diﬀracted SPP wavevectors may be matched with incident plane waves for each polari-

sation case. We do this for an asymmetric zigzag grating with

= 0

= 126 nm the

129

8. Optical Response of Asymmetric Zigzag Bigratings

0.0 0.2 0.4 0.6 0.8 1.0

∗

0 L/2 L

T M

T E

region 1

region 2

Figure 8.6: Magnitude of the electric ﬁeld in region 1 (dashed) and region 2 (dotted)

for the two polarisation cases of TM (black) and TE (red) polarised light, as a function

of oﬀset, δ.

oﬀset which was measured from the SEMs of the produced sample. Since the

T E

and

T M

functions are neither odd nor even, we expand them as a complex Fourier sum,

and the take the magnitude of the Fourier components squared to obtain a qualitative

understanding of the relative scattering eﬃciencies [

116

]. The complex Fourier series

coeﬃcients are given by,

−L

f(x) exp (−inx) dx (8.7)

Where

is the order of the Fourier harmonic. The Fourier components for both

T E

and

T M

functions are plotted in ﬁgure 8.7. The eﬃciency with which light will couple

to the SPPs on an asymmetric zigzag grating will be directly linked to the amplitudes

of these Fourier coeﬃcients, as the strength of each harmonic equates to the scattering

amplitude of the SPPs. These coeﬃcients are diﬀerent for each polarisation, but they

will not be zero.

This leads to the conclusion that both TE and TM polarised light will couple to

all orders of SPPs on such a grating, with diﬀerent (but non-zero) coupling strength

depending on the diﬀracted order used to couple to the SPP. It is possible to excite any

diﬀracted SPP mode with any polarisation of light on such a grating. That is to say,

the same SPP mode may be driven with either polarisation.

130

8. Optical Response of Asymmetric Zigzag Bigratings

0.00 0.02 0.04 0.06 0.08

-4 -3 -2 -1 0 1 2 3 4

(a) TE

0.002 0.006 0.010

-4 -3 -2 -1 0 1 2 3 4

(b) TM

Figure 8.7: The square magnitude of the Fourier coeﬃcients for

= 0

... ±

4 with

an oﬀset of

= 0

, which corresponds to the experimentally measured oﬀset of

126 ± 5 nm for the (a) TE case and (b) TM case.

Incidentally, examining the Fourier coeﬃcients for no oﬀset (setting

= 0), we

recover the polarisation selectivity predicted in chapter 7 for a symmetric zigzag grating.

8.3.2 Results

The coupling of light to SPPs on an asymmetric zigzag grating is demonstrated ex-

perimentally in ﬁgure 8.8. This ﬁgure plots reﬂectivity for both TE and TM po-

larised light illuminating an asymmetric zigzag grating as a function of polar angle

of incidence (

), at

= 0 and

= 500 nm. The reﬂectivity was obtained using

the experimental method outlined in chapter 4, with a robotic table controlling the

polar angle and a spectrometer setting the incident wavelength. The polar angle

range is 7

◦

< θ <

◦

. The zigzag grating sample has the following parameters:

= 597

5 nm

, λ

= 156

9 nm

, d ≈

40 nm

, δ

= 126

5 nm with a zigzag amplitude

of 121 ± 3 nm, measured from the sample SEMs.

Both polarisations show a reﬂectivity minimum at

θ ≈

◦

, characteristic of the light

resonantly driving the

−

scattered SPP. Both curves also show a second resonant

minimum at

θ ≈

◦

, indicative of the light interacting with the +2

SPP. In both

the TE and TM case, these reﬂectivity minima lie at the same angle to within the

experimental error (shown by grey bars in the ﬁgure) for each polarisation, suggesting

it is indeed the same SPP mode being excited in each case.

131

8. Optical Response of Asymmetric Zigzag Bigratings

10 20 30 40 50

0.4 0.5 0.6 0.7 0.8 0.9 1.0

polar angle, θ

−1 +2

Figure 8.8: The TE (black) and TM (red) reﬂectivity of an asymmetric zigzag grating

at a wavelength of

= 500 nm. The blue dotted lines show the calculated position

of the

−

1 and +2 diﬀraction edges. The grey bars are a width equal to the minimum

step-size of the experiment, centred at the average mode minima.

This is an example of a zigzag grating coupling either polarisation to the SPP on the

zigzag surface, as the impinging electric ﬁeld on the asymmetric zigzag proﬁle provides

a normal component with which to induce local surface charge regardless of polarisation.

Since the electric vectors for TE and TM polarised light lie orthogonal to each other,

we may generalize and state that any plane polarisation of light will excite SPPs on

asymmetric zigzag gratings.

The reﬂectivity minimum associated with the +2

scattered SPP at

θ ≈

◦

lower for the TM polarised case than the TE. While the minima associated with

−

shows the opposite, with TM polarised light coupling stronger than the TE polarised

light. Returning to the Fourier coeﬃcients shown in ﬁgure 8.7, this is unsurprising.

These coeﬃcients relate to the strength by which surface ﬁelds may couple to the plane

wave light. With

larger for the TE case, and

larger for the TM case, we

expect the

scattered SPPs to be couple more strongly to the TE polarised light,

and the

scattered SPPs to be couple more strongly to TM polarised light, as

observed in our experimental results.

The dispersion of the in-plane SPPs (

= 0

◦

) propagating along an asymmetric

zigzag grating was measured from the reﬂectivity of TE and TM polarised light, and

these results are shown in ﬁgure 8.9. These data were obtained using the methods

132

8. Optical Response of Asymmetric Zigzag Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4 4.5

× 10

-1

(a) TE

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4 4.5

× 10

-1

(b) TM

Figure 8.9: The dispersion of the SPP modes mapped as a function of (a) TE and (b)

TM reﬂectivity on an asymmetric zigzag grating. The blue lines are the calculated

positions of the diﬀracted light lines. The colour scale indicates the absolute reﬂectivity.

outlined in chapter 4, with the polar angle range of 7

◦

< θ <

◦

in steps of 0

◦

, and a

wavelength range of 400 nm < λ

< 850 nm in steps of 2 nm.

The SPPs associated with

and +2

diﬀraction are clearly coupled to in

each case. In the TE case in ﬁgure 8.9(a), the

SPPs are coupled strongly,

and the +2

scattered SPP is coupled, albeit weaker than the

case. Under

TM polarised illumination, the dispersion shown in ﬁgure 8.9(b) shows the opposite

response, with the stronger coupled mode having scattered by +2

, and the weaker,

yet still coupled mode, having scattered by

. This is consistent with the predicted

magnitude of the Fourier components for each polarisation case, shown in ﬁgure 8.7,

and may be considered a perturbation of the symmetric zigzag case detailed in chapter

7, with the polarisation response of the surface still partially governed by the same

zigzag considerations, but with additional, smaller Fourier components destroying the

symmetry which led to the perfect separation of TE and TM coupled SPPs. The

additional Fourier components in the surface proﬁle have provided direct scattering

routes for the excitation of the +2

scattered SPPs observed for both polarisation

cases in ﬁgure 8.9.

For unpolarised light, the dispersion is mapped in ﬁgure 8.10. The coupling to the

SPP dispersion is now found to be reasonably symmetric around the ﬁrst BZ boundary

(at

≈

−1

), as the SPP eigenmodes of the surface are coupled to by either

the TE or TM components of the unpolarised light. The level of absorption is

≈

40%

with this non-optimised sample. It is possible that a sample that is optimised for the

133

8. Optical Response of Asymmetric Zigzag Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4 4.5

× 10

-1

Figure 8.10: The dispersion of the SPP modes mapped as a function of unpolarised

light reﬂectivity on an asymmetric zigzag grating. The blue lines are the calculated

positions of the diﬀracted light lines. The colour scale indicates the absolute reﬂectivity.

coupling of light to the surface modes could provide 100% absorption of polarised light

and excitation of the SPP bands. This would be a highly eﬃcient mechanism by which

to excite the same SPP mode along a surface, since on ordinary diﬀraction gratings,

50% of un-polarised incident light will not couple at

= 0

◦

. Plasmonic devices such

as SPP enhanced ﬁbre Bragg gratings [

145

146

] could beneﬁt from such an optimised

structure as conversion of TE incident light in to a SPP surface wave with TM character

could improve the coupling into TM ﬁbre optic modes. SPP enhanced photo-detectors

[147] could also beneﬁt from these relaxed polarisation constraints.

8.4 Band Structure of SPPs on an Asymmetric Zigzag

Grating

In chapter 7, we saw how a zigzag grating which possesses mirror symmetry has strict

constraints on the possible band structure of SPPs travelling along such a grating. It

was shown that the interaction of the

−

and +1

scattered SPPs is very weak

and, most strikingly, that the formation of a band-gap at the ﬁrst BZ (the interaction

between the

−

and +2

scattered SPPs) was forbidden by the symmetry of the

charge distributions.

In the case of an asymmetric zigzag grating, both these conditions on the SPP band

structure are no longer true. Band-gaps are experimentally observed at both the high-

134

8. Optical Response of Asymmetric Zigzag Bigratings

(a) Symmetric (b) Asymmetric

Figure 8.11: The iso-frequency contours at

= 550 nm measured on: (a) a symmetric

zigzag from chapter 7; (b) An asymmetric zigzag grating. Along

= 0 two SPP bands

cross in both cases, one scattered from

−k

, and one from +

. The red square inset

in (b) shows the magniﬁed region of the band gap, with the red arrows showing that

the direction of the group velocity contains no k

component.

symmetry points: at

= 0 and at the ﬁrst BZ boundary,

π/λ

. The following

section outlines the experimental observations of these band-gaps on an asymmetric

zigzag grating, and oﬀers a discussion on the origin of these SPP energy gaps. Further

to these results, evidence is found that the combination of highly anisotropic dispersion

of SPPs and the allowed formation of band-gaps may forbid the propagation of SPPs in

all directions for a range of frequencies, forming a full plasmonic band-gap.

8.4.1 Band Gaps at k

= 0

The interaction of the

±k

SPP modes at

= 0 requires a suﬃciently large Bragg

scattering amplitude of the SPPs so that they might interfere strongly and form a

standing wave. The two possible SPP standing waves, which lay along the zigzag

surface and not over the grooves, will possess energetically diﬀerent arrangements of

charge density. These standing waves have diﬀerent frequencies, and between these

two frequencies SPP propagation is forbidden; a plasmonic band gap exists. This

second point is also true of symmetric zigzags, covered in chapter 7, but SPPs on these

symmetric gratings are only weakly Bragg scattered, as the surface functions contains

no signiﬁcant even components to couple the SPP modes together. In the asymmetric

zigzag case, however, the surface function of the asymmetric zigzag pattern introduces

higher harmonics to the grating surface proﬁle, both odd and even. These harmonics

provide a direct coupling mechanism of 2

for the

±k

SPP modes meeting at

= 0,

135

8. Optical Response of Asymmetric Zigzag Bigratings

and consequently a plasmonic band gap may form.

This is observed for an asymmetric silver zigzag grating fabricated as outlined in

section 8.2, using imaging scatterometry to obtain the iso-frequency contours of the SPP

modes. For the case of an illuminating wavelength of

= 550 nm, the obtained contour

is shown in ﬁgure 8.11(b). For comparison, the scattergram of the symmetric zigzag

grating from chapter 7 is also included in ﬁgure 8.11(a). Both experimental maps use the

polarisation which gives the greatest contrast of the contours over the entire momentum

range. Due to SPPs being excited with either polarisation in a asymmetric case, the

iso-frequency contour in ﬁgure 8.11(b) is recorded using unpolarised light, while the

symmetric zigzag is recorded using plane polarisers. These experimental results show

that at the intersection of the two scattered SPP contours at

= 0, the SPP contours

are split in to two distinct bands for the asymmetric case. The group velocity of the

SPP modes is

∇ω

(

), and so the local group velocity direction is determined on

these diagrams as orthogonal to the local contour. At

= 0, there is no component

in the

direction, since in this direction standing waves have formed. Compare

this to the result shown in ﬁgure 8.11(a), for the zigzag grating from chapter 7. These

SPP bands pass through each other unperturbed, having only weak multiple-scattering

processes available to them by which to couple.

The surface of our sample provides the strong scattering and so stronger interaction

of these scattered SPP modes by a combination of a larger zigzag amplitude and the

asymmetery introduced by oﬀsetting an apex by δ = 150 nm.

8.4.2 Band Gaps at the 1

BZ Boundary

The lack of a band-gap at the ﬁrst BZ for SPPs on symmetric zigzag gratings detailed

in chapter 7 is a result of the two possible standing wave solutions being degenerate in

energy. This is because the two possible arrangements for a standing wave of charge

density on a zigzag surface lie in identical electromagnetic environments. Since there is

no diﬀerence in energy between the two states, no band gap is observed.

In the experimentally mapped dispersion of the asymmetric zigzag grating, shown

in ﬁgure 8.10, a clear band-gap has now opened at the ﬁrst BZ, where the

−

and +2

scattered SPPs cross. In this angular frequency range of 3

rad

−1

to 4.2 × 10

rad s

−1

, there is no coupling of light to either SPP, meaning no SPP

propagation occurs within this frequency gap for this orientation of the grating. The

gradient of the SPP contours falls to zero at the BZ, meaning that the SPPs on the

upper and lower frequency band edges have zero group velocity, and are standing waves.

This band gap is a consequence of the broken mirror symmetry of the diﬀraction

grating, and may be demonstrated using a FEM model. In ﬁgure 8.12 we show the

136

8. Optical Response of Asymmetric Zigzag Bigratings

x (nm)

y (nm)

0 300 600

0 75 150

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

5.1 5.15 5.2 5.25 5.3 5.35

× 10

2.8 2.81 2.82 2.83 2.84

(a) Symmetric

x (nm)

y (nm)

0 300 600

0 75 150

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

5.1 5.15 5.2 5.25 5.3 5.35

× 10

2.68 2.69 2.7 2.71

(b) Symmetric

x (nm)

y (nm)

0 300 600

0 75 150

0 150

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

5.1 5.15 5.2 5.25 5.3 5.35

× 10

2.79 2.8 2.81 2.82 2.83 2.84

Figure 8.12: The calculated eigenmodes for the crossing of the

−

and +2

scattered SPPs at the ﬁrst BZ for the cases of: (a) a symmetric zigzag; (b) a symmetric,

high-amplitude zigzag and; (c) an asymmetric zigzag. The unit cell for each is shown

above the modelled dispersion plots.

modelled dispersion for the

−

and +2

SPP crossing at the ﬁrst BZ. The model

used is an eigenmode calculation using the FEM modelling techniques described earlier.

Three dispersions are shown, each for a diﬀerent zigzag grating.

Figure 8.12(a) shows the dispersion at the BZ boundary for a typical symmetric

zigzag grating. As detailed in chapter 7, band-gaps are forbidden to form at this BZ

boundary by the symmetry considerations of the allowed charge distributions. As such,

the SPP dispersion is unperturbed, and the

−

and +2

scattered SPPs pass

through each other without interacting. This is also the case in ﬁgure 8.12(b), where

the symmetric zigzag amplitude has been increased by 150 nm. This unit cell maintains

the mirror symmetry of the zigzag grating and so, once again, formation of a band gap

is forbidden. However, by oﬀsetting the zigzag grating, as modelled in ﬁgure 8.12(c),

the formation of a band gap is immediately achieved in the modelled SPP dispersion.

The group velocity (the gradient of the SPP dispersion curves) falls to zero at the BZ

boundary, and a region of forbidden SPP propagation opens up.

The formation of a band gap at the ﬁrst BZ boundary is a consequence of the SPP

standing waves diﬀering in energy. Figure 8.13 shows the modelled magnitude of electric

ﬁeld plotted in the xz plane for the low and high energy standing wave solutions.

These two standing wave solutions have three ‘hot-spots’ of electromagnetic ﬁeld

137

8. Optical Response of Asymmetric Zigzag Bigratings

x (nm)

z (nm)

0.01

0.02

0.03

0.04

0.05

0.06

0 300 600

0 100 200 300 400

(a) low energy

x (nm)

z (nm)

0.05

0.10

0.15

0.20

0.25

0 300 600

0 100 200 300 400

(b) high energy

Figure 8.13: Field plots of

|E|

(colourplot) a for the (a) low energy and (b) high energy

standing waves at the ﬁrst BZ. The cross section is taken at y = 75 nm.

per unit cell, equating to a wave with a wavevector 3

2, as you would expect for

a SPP standing wave at the ﬁrst BZ formed between the

−

and +2

scattered

SPPs . The diﬀerence in energy between the two modes is easily identiﬁable by the

decay lengths of the electric ﬁelds. In the low energy case, the SPP dispersion is pushed

down in energy, away from the light line. Consequently, the electric ﬁeld proﬁle is more

‘plasmon-like’, conﬁned closer to the surface than the high energy solution, which is

more ‘photon-like’, having been pushed up in frequency, towards the diﬀracted light

line.

The origin of this diﬀerence in energy between the two standing wave states is due

to the diﬀerent arrangements of surface charge along the zigzag. This is demonstrated

in the modelled magnitude of electric ﬁeld plotted in a plane just above the zigzag,

displayed in ﬁgure 8.14.

In this ﬁgure, the two standing wave solutions are shifted spatially by a quarter

phase lag along the grating. This positions the nodes and anti-nodes of the standing

wave as shown in the ﬁgure, with the low energy solution placing the ﬁeld extrema

close to the apexes of the zigzag, and placing the high ﬁeld points for the high energy

solution along the edges of the zigzag. Without any mirror symmetry of the zigzag

grating, these two arrangements for the standing wave surface charge are diﬀerent, and

so an energy diﬀerence exists between them. As a consequence, an SPP band gap now

opens at the ﬁrst BZ boundary.

8.4.2.1 Coupling of Light to the Band Edges

Figure 8.15 shows the experimentally mapped band gap at the ﬁrst BZ boundary for

the three polarisation cases; TE, TM and average polarised light. It is seen that in

138

8. Optical Response of Asymmetric Zigzag Bigratings

x (nm)

y (nm)

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0 300 600

0 75 150

(a) low energy

x (nm)

y (nm)

0.1

0.2

0.3

0.4

0.5

0 300 600

0 75 150

(b) high energy

Figure 8.14: Field plots of

|E|

(colourplot) and

(arrows) for: (a) the low energy and;

(b) high energy standing waves at the ﬁrst BZ. The cross sectional height is 1 nm above

the grooves at 41 nm.

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.2

0.4

0.6

0.8

1.0

0.3 0.4 0.5 0.6 0.7

× 10

3.6 3.8 4 4.2 4.4

× 10

(a) TE

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.2

0.4

0.6

0.8

1.0

0.3 0.4 0.5 0.6 0.7

× 10

3.6 3.8 4 4.2 4.4

× 10

(b) TM

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.2

0.4

0.6

0.8

1.0

0.3 0.4 0.5 0.6 0.7

× 10

3.6 3.8 4 4.2 4.4

× 10

Figure 8.15: Experimental reﬂectivity colour plots mapping the dispersion of the band

gap around the 1st BZ, for incident (a) TE polarised light, (b) TM polarised light and

all three cases, the lower energy band edge is coupled to by both TE and TM light

relatively strongly with a dark band of

≈

20% reﬂectivity, while the higher-frequency

band edge is never coupled to as strongly, with a reﬂectivity of

≈

90%, absorbing only

≈ 10% of the incident light.

To determine why light will preferentially couple to the lower frequency mode, we

may consider the overlap integral of the ﬁelds for both cases. The overlap integral

gives a quantitive measure of how much a driving electric ﬁeld will match that of the

resonance, and hence allow us to determine relative coupling strength between the two

modes. The overlap integral is deﬁned in a plane as [148],

ζ =

dA|

(8.8)

139

8. Optical Response of Asymmetric Zigzag Bigratings

and

are the two electric ﬁelds to consider, in our case the SPP eigenmode and

the incident plane wave. A plane of area

integrated over an area equal to the plane of

incidence, in this case (

= 0), the

plane. We shall use ﬁeld proﬁles for the resonant

standing waves from the calculated FEM model and make several simpliﬁcations to the

overlap expression for

. Firstly, we note that the

dependence for the electric ﬁeld

of a SPP has the same functional form in each case, an exponential in the

direction.

Immediately then, we can simplify the integral to a one-dimensional line integral in

as the normalized contributions from the

of the SPP and the light ﬁeld overlap will

be equal in both cases. Secondly, we take the sum of the

ﬁeld along

to account for

the whole unit cell. Since this is now a numerical calculation, the expression for the

simpliﬁed overlap, ζ

is,

x=0

SP P

x=0

SP P

x=0

(8.9)

The data for

SP P

(

) are extracted from the numerical simulation shown in ﬁgure

8.14, for a region in the grooves at

= 20 nm. At the ﬁrst BZ boundary, the incident

light,

(

) has an in-plane wavevector equal to

2, and so the expression used

for E

is,

= sin



+ ψ



(8.10)

with

equal to the phase of the wave at the surface. Since the band-gap under

consideration is not for normal incidence, the incident electric ﬁeld propagates in the

direction, which we can represent as a change in

. At certain values of

, optimal

coupling will occur and at 90

◦

in phase to this, no coupling will occur. When considering

the overlap integral for each standing wave case, we optimise the value of

to give the

largest value of

, corresponding to a phase at which the light is optimally matched to

the standing-wave ﬁelds. The functional forms for both

and

SP P

, for both band

edge solutions are shown in ﬁgure 8.16.

It is found that for the low energy solution,

−

= 0

0212(4

d.p.

), while for the high

energy solution

= 0

0000(4

d.p.

). The overlap of the impinging

is far greater for

the lower energy solution by comparison with the high energy solution, which to a fair

approximation is zero. Returning to ﬁgure 8.16, the physical reason for this preferred

coupling is now clear. In the low energy case,

−

is compressed (and so enhanced) in

region 2 (450 nm

< x <

600 nm). This region correspond to

originating from just

right of the zig-zag apex crossing the grooves to the lower apex edge at

= 600 nm,

whereas for

, the ﬁeld is reasonably uniformly distributed across the zigzag. The

asymmetrically enhanced ﬁeld in the

−

case adds a

2 component to the SPP ﬁeld,

140

8. Optical Response of Asymmetric Zigzag Bigratings

-1.0 -0.5 0.0 0.5 1.0

x (m)

(a.u.)

0 2 × 10

-7

4 × 10

-7

6 × 10

-7

8 × 10

-7

1 × 10

-6

1.2 × 10

-6

(a) low energy

-1.0 -0.5 0.0 0.5 1.0

x (m)

(a.u.)

0 2 × 10

-7

4 × 10

-7

6 × 10

-7

8 × 10

-7

1 × 10

-6

1.2 × 10

-6

(b) high energy

Figure 8.16: The (a) low energy and (b) high energy

-component of the SPP standing

wave electric ﬁeld (black,

SP P

) and incident ﬁeld (red,

) varying across two unit

cells of the asymmetric zigzag grating (light grey shapes indicate the zigzag in the

plane).

allowing it to couple strongly to the light, which itself has a

2 component at the BZ

boundary. Similar explanations exist for the coupling for band edges on surface relief

blazed gratings [85, 114] explained in chapter 4.

8.4.2.2 A Full Surface Plasmon Band Gap

SPP propagation on a zigzag grating is naturally anisotropic with respect to propagation

direction, due to the large band gaps which can form when the SPPs run over the

grooves, and the relativity small (or forbidden) band gaps that form when the SPPs

run along the zigzags. In chapter 7, it was shown that this anisotropy can be so large

that a zigzag grating may form the basis of a SPP collimating device, with the group

velocity of SPP modes solely in one direction. The asymmetric zigzag also shows this

increasing anisotropy with frequency, as shown in ﬁgure 8.17. As the frequency of the

light is increased, the SPP contours lie further from the zero-order circular SPP cones,

and become more ellipsoidal. These contours also split due to band-gaps along the

= 0 symmetry plane.

The anisotropic propagation combined with the ability for SPPs to form band gaps

at the 1

BZ boundary lead to a mechanism by which we may forbid the propagation

of SPPs in all directions along the grating surface, forming a full plasmonic band gap.

When the dispersion of the SPPs has become suﬃciently anisotropic so that the

group velocity of the SPPs are in a single direction, the SPP contours intersect the

ﬁrst BZ at

π/λ

with momentum only in the

direction. This was observed before

in the symmetric zigzag case in chapter 7: ﬁgure 7.23, where the SPP contours pass

141

8. Optical Response of Asymmetric Zigzag Bigratings

(a) 650 nm (b) 600 nm (c) 580 nm

(d) 550 nm (e) 500 nm (f) 450 nm

Figure 8.17: Iso-frequency contours of an asymmetric zigzag grating for a range of wavelengths, measured using the reﬂectivity of

unpolarised light in the imaging scatterometer. The blue circles indicate calculated diﬀraction edges.

142

8. Optical Response of Asymmetric Zigzag Bigratings

in-plane wavevector, k

−1

)

angular frequency, ω (rad s

−1

)

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1 1.2

× 10

2.5 3 3.5 4 4.5

× 10

-1

(a) (b)

Figure 8.18: (a) The dispersion of the SPP modes on an asymmetric zigzag grating for

= 0

◦

mapped using the reﬂectivity of unpolarised light. Included are the wavelength

slices corresponding to the iso-frequency scattergrams in ﬁgure 8.17, ranging from low

frequency (red, 650 nm) to high (blue, 450 nm ), (b) The scattergram for

= 450 nm,

corresponding to the highest frequency slice in (a).

through the BZ boundary with the contour parallel to the boundary. In the asymmetric

zigzag grating case, the SPP iso-frequency contours still approach the BZ boundary

approximately parallel, but now form a band gap at the boundary. This means that the

SPP band gap at this boundary forms at all values of momentum simultaneously, and

so a frequency band over which all SPP propagation is forbidden is opened. This is a

full plasmonic band gap. Figure 8.18 shows an iso-frequency contour for

= 450 nm

close to the forbidden frequency region, and shows minimal coupling for light to SPPs

across the entire light cone. This iso-frequency contour is not at the exact frequency of

the observed band gap, shown in the corresponding dispersion plot in ﬁgure 8.18, due

to experimental limitations. The poorly coupled SPP contours in the scattergram also

show stronger (yet still weak) coupling towards the edges of momentum-space, and it is

clear that the contours are, in fact, slightly curved. This shows that the dispersion is

not as anisotropic as would be necessary for the contour to intersect the BZ boundary

simultaneously. However, these results demonstrate in principle the engineering of a full

plasmonic band gap by using a sub-wavelength periodicity to introduce large anisotropy

in SPP propagation and then the ability for such gratings to form band gaps at the BZ

in all directions simultaneously. Optimising the sample and experimentally measuring

the forbidden frequency band is left as future work.

143

8. Optical Response of Asymmetric Zigzag Bigratings

8.5 Conclusions

We have demonstrated in this chapter that manipulation of the surface symmetry on a

zigzag grating can lead to some novel optical eﬀects.

The coupling of any polarisation to SPP modes on such gratings was explained and

experimentally veriﬁed in section 8.3, with both TE and TM polarised light resonantly

driving the same SPP modes. The coupling strength was still preferentially to TE or

TM polarised light depending on the order of diﬀraction involved, but the optimisation

of this coupling might be possible using a diﬀerent broken symmetry geometry.

The formation of band gaps on these gratings has also been measured, for both

planes of symmetry accessible to experiment for this sample. The energy diﬀerence

between these standing waves arises from the energetically dissimilar arrangements of

charge along the asymmetric zigzag. The coupling of light to these standing wave states

is found to depend on the symmetry of the surface, analogous to how the phase of

Fourier components determines the light coupling on surface-relief gratings. This is

explained in depth in section 8.4.

Finally, the potential for this structure to provide a full surface plasmon band gap

is explored, with the combination of the highly anisotropic SPP dispersion and the

formation of band-gaps at the BZ boundary showing promise for the engineering of a

full surface plasmon band-gap device.

144

Chapter 9

Conclusions

9.1 Summary of Thesis

This thesis details original experimental investigations on the excitation of SPPs along

metallic diﬀraction grating surfaces. Broadly, these investigations can be divided in to

two main categories: the investigation of SPPs on traditional ‘crossed’ bigratings with

novel symmetries, and the introduction of a new type of diﬀraction grating supporting

SPPs, the ‘zigzag’ grating.

The work on the traditional ‘crossed gratings’ is presented in chapters 5 & 6, exploring

gratings with symmetries that have previously received little attention in the literature.

Chapter 5 experimentally explores the dispersion and coupling of SPPs on a rectangular

bigrating. The groove proﬁles of such bigratings are described as a Fourier expansion

of the constituent gratings, and this is used to explain the experimentally observed

scattering and interaction of the supported SPP modes. Surface plasmon band-gaps

are seen to occur at the boundaries of the rectangular BZ and are recorded using the

novel technique of imaging scatterometry. This measurement technique for acquiring

the iso-frequency contours of SPPs in

-space is original to this thesis, and detailed in

chapter 4. The measured SPP contours at the BZ boundary are reproduced using FEM

modelling, showing good agreement with experiment.

Chapter 5 also shows that by increasing the depth of a constituent grating, the SPP

travelling in the orthogonal direction can be slowed in group velocity. Experimental

results also show that the SPP iso-frequency contour can be shaped by changing the

available grating harmonics and hence the strength of scattering amplitudes. This

is demonstrated experimentally as a mechanism by which to design anisotropic SPP

propagation, which could be useful for the development of SPP surface optics.

Chapter 6 investigates SPPs on gratings with the lowest symmetry of all the 2D

Bravais lattices, oblique bigratings. SPP mediated polarisation conversion is observed

145

9. Conclusions

on these gratings due to the broken mirror symmetry of the surface. Measurements of

the dispersion also show evidence that SPPs may undergo self-coupling. This is the

mechanism of a propagating SPP mode being able to resonantly drive another, diﬀerent,

SPP mode. This is a possibility providing the two modes do not travel orthogonally to

one another.

The iso-frequency contours of these SPPs are mapped experimentally using imaging

scatterometry, and it is found that band-gaps do not necessarily occur at the conventional

deﬁnition of the BZ boundary. A discussion of the symmetry considerations on such a

grating is used to explain why this is the case, concluding that since the BZ boundary

is itself not a contour of high-symmetry for an oblique lattice, there is no condition

for the band-gaps to form at this arbitrary boundary. There are, however, isolated

high-symmetry points along the BZ boundary, and when the plane of incidence intersects

these unique points, it is experimentally observed that SPP band-gaps do still occur.

The ﬁnal two experimental chapters introduce a new type of diﬀraction grating: the

zigzag grating. These are gratings where a diﬀractive periodicity has been introduced

by ‘zigzagging’ a set of sub-wavelength (non-diﬀracting) grooves in a metal surface. The

electric ﬁeld of plane polarised light incident on such a zigzag grating will intersect the

surface regardless of the polarisation angle, inducing surface charge. This results in

SPP excitation on these gratings with either TE or TM polarised light.

Chapter 7 details such a zigzag grating that possesses a single mirror plane. The

symmetry of this zigzag structure leads to the observation that TE polarised light

excites SPPs scattered by odd-order grating vectors, while TM polarised light excites

SPPs scattered by even-orders. This separation of SPP diﬀracted orders by polari-

sation selectivity is explained using a simple theoretical treatment, and is observed

experimentally.

A second consequence of this symmetric zigzag grating is that the standing surface

wave states at the ﬁrst BZ boundary are found to be degenerate in energy. By modelling

the system using the FEM, it is found that the two possible SPP standing waves inhabit

identical electromagnetic environments, and so no energy diﬀerence exists between them

and no bad-gap may form. Experimental observations conﬁrm this lack of SPP band-gap

at the ﬁrst BZ boundary.

The ﬁnal results of chapter 7 show that SPP band-gaps associated with the short

sub-wavelength pitch cause the observed SPP iso-frequency contours to deform. So

large is this perturbation that at high frequencies the SPP contours are ﬂat, and the

SPP’s group velocity is constrained to be parallel to the long-pitch direction, irrespective

of incident angle of the coupling light. This, combined with the forbidden band-gaps,

makes these zigzag gratings excellent candidates for surface wave collimation devices.

The ﬁnal experimental results of this thesis are shown chapter 8 and pertain to the

146

9. Conclusions

SPPs excited along a zigzag grating which possesses no mirror symmetry. This leads

to the experimental observation that either TE or TM polarised light may excite the

same SPP modes. Band-gaps may now form, as the asymmetric zigzag surface leads

to diﬀerent energetic arrangements of surface charge for the diﬀerent standing wave

states. The anisotropic propagation of SPPs on such a surface observed in chapter 7

is also observed for these asymmetric zigzag gratings, which, when combined with the

large SPP band-gaps, leads to the formation of a full SPP band-gap, for which SPP

propagation is forbidden in all directions.

9.2 Future Work

Throughout this thesis we have demonstrated some novel optical eﬀects related to SPPs

on metal gratings, and this provides a wealth of possible future work.

The manipulation of SPP band structures presented using rectangular bigratings

in chapters 5, 7 and 8 may provide useful tools in the design of SPP surface optics,

including lenses, collimators and perhaps even negative index devices. Smoothly varying

the depths along such gratings would change the eﬀective mode index along the surface,

and be used to guide or direct SPPs.

The use of sub-wavelength grooves shaped along their length to provide diﬀractive

coupling to SPPs has a large potential for many interesting future studies. The use of

the grating in chapter 7 as a collimating device for surface waves would be an excellent

avenue for investigation. A schematic of a possible experimental arrangement is shown

in ﬁgure 9.1. In this ﬁgure, a SPP point source (red dot) excites and couples to SPPs

on a planar metal surface. The use of a near-ﬁeld optical microscopy (SNOM) tip or

a small point scatterer illuminated with a laser could provide near-ﬁeld coupling into

a SPP wave which, in the absence of the grating, will propagate outwards radially.

With the addition of a grating similar to the one presented in chapter 7, the allowed

momentum states of the SPP are constrained to propagate in a single direction only,

causing collimation of the surface wave. The distance between the grating and the

point source would control the ﬁnal (presumably Gaussian) SPP beam width. This

collimation could then be observed by measuring the electric ﬁeld across the surface

using SNOM, or by using leaky radiation microscopy [61] to image the beam.

The potential exists, by altering the zigzag period, the sub-wavelength period or the

depth of the sub-wavelength grooves, to allow for the onset of ﬂat SPP contours at a

lower frequency than that found in chapter 7, possibly providing a device which can

collimate SPP waves over a broad range of frequencies.

Another potential device that could incorporate a zigzag diﬀraction grating is an

SPP mediated light source. Metal-Insulator-Metal (MIM) devices with the top metal

147

9. Conclusions

SPP Source

SP P

Zigzag grating

Figure 9.1: A possible experimental arrangement for the observation of collimated

surface waves using zigzag gratings.

surface corrugated to form a diﬀraction grating have been shown to emit light when

driven with a suitable voltage [

149

]. This process is mediated by the excitation of SPPs

by electrons followed by the out-coupling of these SPPs to light by the diﬀraction grating.

Replacing the top surface diﬀraction grating with the symmetric zigzag grating described

in chapter 7 would result in the emission of TE polarised light into the

, ±

, ...

diﬀracted orders and also the emission of TM polarised light into the

, ±

, ...

orders.

The device would be a polarisation directing/separating light source. The use of an

asymmetric zigzag as detailed in chapter 8 would result in the emission of unpolarised

light into the diﬀracted orders.

The asymmetric zigzag in chapter 8 also provides some exciting possibilities for other

future work. Investigation into the eﬀect of diﬀerent asymmetric patterns would most

likely lead to eﬀects analogous to changing the phase of a 2

component on a traditional

surface-relief grating. If a device could be constructed where the unpolarised light could

couple to the SPP standing wave states (and so a region of high density of SPP states)

in the upper band edge (as opposed to the preferential coupling to the lower band

edge observed in chapter 8), one could envisage a device with great potential in SPP

enhanced lasers or SPP enhanced solar cells. In such devices, it is desirable to couple

the SPP into a gain medium coating the surface, which favours the upper-band edge

of SPP band-gaps since these ﬁelds are more ‘photon-like’ extending further into the

surround gain medium that the ‘plasmon-like’ lower-band edge. The advantage of using

an optimised asymmetric zigzag is that the same SPP modes can be coupled to with

any incident polarisation, possibly improving eﬃciency. The polarisation insensitivity

148

9. Conclusions

to the generation of the locally TM polarised SPP waves could also prove useful for the

eﬃcient coupling of light into optical ﬁbres.

149

Publications

Journal Articles in Print

Surface Plasmons on Zig-Zag Gratings, T. J. Constant, T. Taphouse, H. J. Rance,

S. C. Kitson, A. P. Hibbins, and J. R. Sambles. Opt. Express

, 23921–23926

(2012).

Surface Waves at Microwave Frequencies Excited on a Zigzag Metasurface, H.

J. Rance, T. J. Constant, A. P. Hibbins, and J. R. Sambles. Phys. Rev. B

125144 (2012).

Journal Articles in Preparation

Mapping Surface Plasmon Iso-Frequency Contours Using Imaging Scatterometry,

T. J. Constant, A. P. Hibbins, A. P. Lethbridge, E. K. Stone, J. R. Sambles and

P. Vukusic. (2013).

Poster Presentations

Direct Imaging of Surface Plasmon Band-gaps on zig-zag gratings, T. J. Con-

stant, A. P. Hibbins and J. R. Sambles. 4

International Topical Meeting on

Nanophotonics and Metamaterials, Seeﬁeld, Tirol, Austria. (January 2013)

Zig-Zag Gratings, T. J. Constant, A. P. Hibbins and J. R. Sambles. 3

Inter-

national Topical Meeting on Nanophotonics and Metamaterials, Seefeld, Tirol,

Austria. (January 2011)

Surface Plasmons on Rectangular Bi-Gratings, T. J. Constant, A. P. Hibbins and

J. R. Sambles. The Institute of Physics: Plasmonics UK Meeting, Institute of

Physics, London, United Kingdom. (May 2010)

150

9. Conclusions

Surface Plasmons on Rectangular Bi-Gratings, T. J. Constant, A. P. Hibbins and J.

R. Sambles. The Royal Society, Theo Murphy International Scientiﬁc Meeting on

Metallic Metamaterials and Plasmonics, Chicheley Hall, United Kingdom. (June

2010)

Future Publications

Other publications arising from this thesis and intended for submission in the near future

include: a paper on the self-collimation characteristics of zigzag gratings presented in

chapter 7, and a paper on the excitation of SPPs on the asymmetric zigzag grating

geometry in chapter 8.

151

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